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Stewart
Consider the vector field
\(\bm{F}\colon\mathbf{R}^2\to\mathbf{R}^2\)
defined as
\(\bm{F}(x,y) = -y\mathbf{i}+x\mathbf{j}.\)
Sketch some of the vectors of this vector field
until you have a good idea of what it looks like.
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Stewart
Sketch the vector field
\(\bm{F}\colon\mathbf{R}^3\to\mathbf{R}^3\)
defined as \(\bm{F}(x,y,z) = z\mathbf{k}.\)
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Stewart
Sketch the gradient field and a contour plot
of the function \(f\) defined as \(f(x,y) = x^2-y^3\)
and notice how they’re related.
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Sketch the vector field
\(\bm{F}\colon\mathbf{R}^3\to\mathbf{R}^3\)
defined as
\(\bm{F}(x,y,z) = 2x\mathbf{i} -2y\mathbf{j} -2x\mathbf{k}.\)
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Sketch the gradient field and a contour plot
of the function \(f\) defined as \(f(x,y) = x^2\sin(5y)\)
and notice how they’re related.
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Sketch the gradient field
of the function \(f\) defined as \(f(x,y,z) = z\mathrm{e}^{-xy}.\)
Review Before Next Class
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What is the length of the parametrically-defined curve
\(\bm{r}(t) = \langle t^2,9t,4t^{3/2} \rangle\)
for \(t\) between one and four?