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  1. Find the gradient \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\del f of the following scalar fields:

    (a) \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}f(\vv x) = x\sin(y) + zy\cos(x)

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\del f =\vc{(\sin(y) - zy\sin(x))}{+(x\cos(y) + z\cos(x))}{+y\cos(x)}

    (b) \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}f(\vv x) = r = \cartr

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\del f = \vc{\frac{2x}{2\cartr}}{+\frac{2y}{2\cartr}}{+\frac{2z}{2\cartr}}

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}~~~~= \vc{\frac{x}{\cartr}}{+\frac{y}{\cartr}}{+\frac{z}{\cartr}}

    (c) \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}f(\vv x) = \ln(r) = \ln(\cartr)

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\del f = \vc{\frac{x}{\cartrsq}}{+\frac{y}{\cartrsq}}{+\frac{z}{\cartrsq}}

  2. Find the divergence \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\del\cdot\vv F and the curl \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\del\times\vv F for the following vector fields:

    (a) \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\vv F(\vv x) = \vc{x^2yz}{+xy^2z}{+xyz}

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\del\cdot\vv F = 2xyz + 2xyz + xy
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}~~~~= 4xyz + xy

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\del\times\vv F = \vc{(\hpder yF_z - \hpder zF_y)}{+(\hpder zF_x - \hpder xF_z)}{+(\hpder xF_y - \hpder yF_x)}
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}~~~~= \vc{(xz - xy^2)}{+(x^2y - yz)}{+(y^2z - x^2z)}

    (b) \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\vv F(\vv x) = \vc{\frac{x}{\cartr}}{+\frac{y}{\cartr}}{+\frac{z}{\cartr}}

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\del\cdot\vv F = \frac{\cartr - x/\cartr}{\cartrsq} + \frac{\cartr - y/\cartr}{\cartrsq} + \frac{\cartr - z/\cartr}{\cartrsq}

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}~~~~= \frac1{\cartr} - \frac{x + y + z}{(\cartrsq)^{3/2}}

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\del\times\vv F = \vc{(\frac{-2yz}{2\cartr} - \frac{-2zy}{2\cartr})}{+(\frac{-2zx}{2\cartr} - \frac{-2xz}{2\cartr})}{+(\frac{-2xz}{2\cartr} - \frac{-2zx}{2\cartr})}

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}~~~~= \vv 0

  3. Maxwell’s equations in vacuum are given by \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\del\cdot\vv E = 0\newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}} \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\del\cdot\vv B = 0\newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}} \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\del\times\vv E = -\frac1{c}\pder{\vv B}{t}\newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}} \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\del\times\vv B = \frac1{c}\pder{\vv E}{t}\newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}

    where \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\vv E = \vv E(\vv x,t) is the electric field, \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\vv B = \vv B(\vv x,t) is the magnetic field, and \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}c is the speed of light in vacuum.

    (a) Using the identities from section 19, show that \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\del\times(\del\times\vv E) = -\frac1{c^2}\pder{^2\vv E}{t^2} and \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\del\times(\del\times\vv B) = -\frac1{c^2}\pder{^2\vv B}{t^2}.

    (b) Use this fact to deduce that \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\vv E and \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\vv B each satisfy the wave equation, hence both the electric and magnetic fields propogate through the vacuum as waves with constant speed \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}c (this is called electromagnetic radiation). That is, show \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\nabla^2\vv E = \frac1{c^2}\pder{^2\vv E}{t^2} and \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\nabla^2\vv B = \frac1{c^2}\pder{^2\vv B}{t^2}

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\del\times(\del\times\vv E) = \del(\del\cdot\vv E) - \nabla^2\vv E
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}-\frac1{c^2}\pder{^2\vv E}{t^2} = \del(0) - \nabla^2\vv E
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\nabla^2\vv E = \vv0 + \frac1{c^2}\pder{^2\vv E}{t^2}
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\nabla^2\vv E = \frac1{c^2}\pder{^2\vv E}{t^2}

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\del\times(\del\times\vv B) = \del(\del\cdot\vv B) - \nabla^2\vv B
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}-\frac1{c^2}\pder{^2\vv B}{t^2} = \del(0) - \nabla^2\vv B
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\nabla^2\vv B = \vv0 + \frac1{c^2}\pder{^2\vv B}{t^2}
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\nabla^2\vv B = \frac1{c^2}\pder{^2\vv B}{t^2}

  4. Find the arclength of the parabola \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}y=x^2 for \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}0\le x\le 1.

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\vv x(t) = \vc{t}{+t^2}{+0},~~ 0\le t\le 1

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\int_0^1\sqrt{1^2 + (2t)^2 + 0^2}dt
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}~~~~= \int_0^1\sqrt{1 + 4t^2}dt

  5. Find the arclength of the helix \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\vv x(t) = \vc{\cos(t)}{+\sin(t)}{+t} for \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}0\le t\le 2\pi

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\int_0^{2\pi}\sqrt{\sin^2(t) + \cos^2(t) + 1^2}dt
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}~~~~= \int_0^{2\pi}\sqrt{1 + 1}dt
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}~~~~= 2\pi\sqrt2

  6. Let \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}f(\vv x) = xyz. Evaluate \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\int_C f\:ds where \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}C is the curve with parametric equation \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\vv x(t) = \vc{2\sin(t)}{+t}{-2\cos(t)}, \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}0\le t\le \pi.

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\int_0^\pi 4t\sin(t)\cos(t)\sqrt{4\cos^2(t) + 1^2 + 4\sin^2(t)}dt
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}~~~~= 4\sqrt5\int_0^\pi t\sin(t)\cos(t)dt
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}~~~~= 4\sqrt5([t(\cos^2(t) - \sin^2(t))|_{t=0}^\pi - \int_0^\pi (\cos^2(t) - \sin^2(t))dt)

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}~~~~= 4\sqrt5(\pi(\cos^2(\pi) - \sin^2(\pi)) - \int_0^\pi \cos(2t)dt)
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}~~~~= 4\sqrt5(\pi - \frac1{2}[\sin(2t)|_{t=0}^\pi)

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}~~~~= 4\sqrt5(\pi - \frac1{2}(0-0))
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}~~~~= 4\pi\sqrt5

  7. Let \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\vv F(\vv x) = \vc{-kx}{-ky}{+0}, where \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}k is constant. Evaluate \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\int_C \vv F\cdot d\vv x for the following curves \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}C:

    (a) \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}(1,1)\to(4,1)\to(4,4) with each arrow representing a straight line connecting the given points in the direction of the arrow

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\text{I}: \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}(1,1)\to(4,1);~~\vv x(t) = \vc{t}{+}{+0},~~1\le t\le 4
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\text{II}: \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}(4,1)\to(4,4);~~\vv y(t) = \vc{4}{+t}{+0},~~1\le t\le 4

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\int_\text{I} \vv F\cdot d\vv x + \int_\text{II} \vv F\cdot d\vv y
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}~~~~= \int_1^4 -ktdt + \int_1^4 -ktdt
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}~~~~= 2(-k\frac1{2}[t^2|_{t=1}^4)

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}~~~~= k(16-1)
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}~~~~= 15k

    (b) \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}(1,1)\to(1,4)\to(4,4) with each arrow representing a straight line connecting the given points in the direction of the arrow

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\text{I}: \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}(1,1)\to(1,4);~~\vv x(t) = \vc{}{+t}{+0},~~1\le t\le 4
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\text{II}: \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}(1,4)\to(4,4);~~\vv y(t) = \vc{t}{+4}{+0},~~1\le t\le 4

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\int_\text{I} \vv F\cdot d\vv x + \int_\text{II} \vv F\cdot d\vv y
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}~~~~= \int_1^4 -ktdt + \int_1^4 -ktdt
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}~~~~= 15k

    (c) the unit circle counterclockwise from angle \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}0 to \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\pi

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\vv x(t) = \vc{\cos(t)}{+\sin(t)}{+0},~~ 0\le t\le \pi

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\int_0^t k\cos(t)\sin(t)dt + \int_0^\pi -k\sin(t)\cos(t)dt
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}~~~~= 0

  8. Let \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\vv F(\vv x) = \vc{\frac{x}{\sqrt{x^2+y^2}}}{+\frac{y}{\sqrt{x^2+y^2}}}{+0}. Evaluate \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\int_C \vv F\cdot d\vv x where C is the parabola given by \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}y=x^2+1 from \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}(-1,2) to \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}(1,2).

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\vv x(t) = \vc{t}{+(t^2+1)}{+0},~~ -1\le t\le 1

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\int_{-1}^1 \frac{t}{\sqrt{t^2+(t^2+1)^2}}dt + \int_{-1}^1 \frac{t(t^2+1)}{\sqrt{t^2+(t^2+1)^2}}dt

  9. Determine whether or not the following vector fields are conservative. If they are, find a scalar potential function for them.

    (a) \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\vv F(\vv x) = \vc{(2x-3y)}{+(-3x+4y-8)}{+0}

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\del\times\vv F = \vc{0}{+0}{+((-3)-(-3))} = \vv0~~~~\cmark

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\hpder x\phi = 2x - 3y
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\phi = x^2 - 3xy + g(y)

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\hpder y\phi = -3x + 4y - 8
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\phi = -3xy + 2y^2 - 8y + h(x)

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\phi = x^2 - 3xy + 2y^2 - 8y

    (b) \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\vv F(\vv x) = \vc{yz}{+xz}{+(xy+2z)}

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\del\times\vv F = \vc{(x - x)}{+(y - y)}{+(z - z)} = \vv0~~~~\cmark

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\hpder x\phi = yz
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\phi = xyz + g_1(y) + g_2(z)

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\hpder y\phi = xz
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\phi = xyz + g_3(x) + g_4(z)

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\hpder z\phi = xy + 2z
    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\phi = xyz + z^2 + g_5(x) + g_6(y)

    \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}}\newcommand{\spder}[3]{\frac{\partial^2#1}{\partial#2\partial#3}}\newcommand{\hpder}[1]{\partial_{#1}}\newcommand{\cmark}{\checkmark}\newcommand{\xmark}{\Chi}\newcommand{\vv}[1]{\vec{\mathbf{#1}}}\newcommand{\vh}[1]{\hat{\mathbf{#1}}}\newcommand{\vc}[3]{#1\vh{i}#2\vh{j}#3\vh{k}}\newcommand{\del}{\vec{\nabla}}\newcommand{\cartrsq}{{x^2 + y^2 + z^2}}\newcommand{\cartr}{\sqrt{\cartrsq}}\phi = xyz + z^2


© Emberlynn McKinney