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Stewart
For \(\bm{F}(x,y,z) = xz\mathbf{i} + xyz\mathbf{j} -y^2\mathbf{k}\) compute the curl of \(\bm{F}.\) -
Stewart
Show that the vector field \(\bm{F}\) defined as \(\bm{F}(x,y,z) = xz\mathbf{i} + xyz\mathbf{j} -y^2\mathbf{k}\) is not conservative. -
Stewart
Show that the vector field \(\bm{F}\) defined as \(\bm{F}(x,y,z) = y^2 z^3\mathbf{i} + 2xyz^3 \mathbf{j} +3xy^2z^2\mathbf{k}\) is a conservative vector field, and find an example of a function \(f\) such that \(\bm{F} = \nabla f.\) -
Stewart
For \(\bm{F}(x,y,z) = xz\mathbf{i} + xyz\mathbf{j} -y^2\mathbf{k}\) compute the divergence of \(\bm{F}.\) -
Stewart
Show that the vector field \(\bm{F}\) defined as \(\bm{F}(x,y,z) = xz\mathbf{i} + xyz\mathbf{j} -y^2\mathbf{k}\) cannot be written as the curl of another vector field. I.e. there is no vector field \(\bm{G}\) such that \(\bm{F} = \operatorname{curl}\bm{G}.\)
Review Before Next Class
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Recall what all of the following parametrically-defined curves look like.
\(\displaystyle \big\langle t^3+1, t \big\rangle\) for \(-1 \lt t \lt 1\)\(\displaystyle \big\langle \cos(t), \sin(t) \big\rangle\) for \(-\pi/3 \lt t \lt 3\)\(\displaystyle \big\langle \cos(t), \sin(2t) \big\rangle\) for \(0 \lt t \lt 2\pi\)\(\displaystyle \big\langle \mathrm{e}^t, \sin(t) \big\rangle\) for \(0 \lt t \lt \mathrm{e}\)
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Recall what all of the following surfaces look like:
\(\displaystyle x^2+y^2+z^2 = 49\)\(\displaystyle x^2+y^2+z = 49\)\(\displaystyle x^2+y^2-z = 49\)\(\displaystyle x^2+y^2-z = 0\)\(\displaystyle x^2-y^2-z = 0\)\(\displaystyle x^2-y^2-z^2 = 49\)\(\displaystyle x^2+y^2-z^2 = 49\)\(\displaystyle x^2+y+z = 49\)\(\displaystyle x+y+z = 49\)\(\displaystyle x+y = 49\)\(\displaystyle y = 49\)