Section 11.3 Assignment 3
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Explain how vector fields might be useful in each of the following situations. (6)
Planning a wind farm.
Shielding a solenoid.
Tracking shoals of fish.
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Draw these vector fields. (You can check this with a computer, but I want to see some of the calculations behind your drawing.)
- \begin{equation*} F(x,y) = (x^2 - y, y^2 -x) \end{equation*}
- \begin{equation*} F(x,y) = \left( \frac{1}{x+y}, \frac{1}{x-y} \right) \end{equation*}
- \begin{equation*} F(x,y) = (x \cos (\pi y), y \cos (\pi x)) \end{equation*}
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Calculate the integral curves for the following vector fields. (8)
- \begin{equation*} F(x,y) = \left( \frac{x}{2}, \frac{y}{4} \right) \end{equation*}
- \begin{equation*} F(x,y) = \left( \frac{1}{x^2}, \frac{1}{y^3} \right) \end{equation*}
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What do integral curves mean for the following vector fields? (9)
The vector field of ocean currents. (2D or 3D)
The vector field of gravitational force per unit mass around a gravitational source. (3D)
The vector field of temperature gradients over a landscape. (2D)
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Prove the follwoing operator composition identities for \(f\) a \(C^2\) scalar field on \(\RR^3\) and \(F\) a \(C^2\) vector field on \(\RR^3\text{.}\) (8)
- \begin{equation*} \nabla \times (\nabla f) = (0,0,0) \end{equation*}
- \begin{equation*} \nabla \cdot (\nabla \times F) = 0 \end{equation*}
If \(F = (F_1,F_2,F_3)\) is a \(C^2\) vector field on \(\RR^3\text{,}\) show that the property of being irrotational is not reserved if the components of \(F\) are re-arranged. (Use an example to show how the property is lost.) Why is the order of the components important? (4)
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Calculate the line integrals.
\begin{equation*} \int_\gamma F \cdot ds \end{equation*}for these \(F\) and \(\gamma\text{.}\) (16)
- \begin{align*} \amp F(x,y) = (x^2 + y^2, x - y + 3) \amp \amp \\ \amp \gamma(t) = (t+1,t-2) \amp \amp t \in [1,5[ \end{align*}
- \begin{align*} \amp F(x,y) = (x, y) \amp \amp \\ \amp \gamma(t) = (3 \cos 2t, 3 \sin t) \amp \amp t \in [0, \pi] \end{align*}
- \begin{align*} \amp F(x,y) = (-4x, -4y) \amp \amp \\ \amp \gamma(t) = (t \cos t, t \sin t) \amp \amp t \in [0, 2\pi] \end{align*}
- \begin{align*} \amp F(x,y) = \left(0, \frac{-1}{x} \right) \amp \amp \\ \amp \gamma(t) = (t^2, t^3) \amp \amp t \in [0, 8] \end{align*}
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Let \(\gamma\) be any path from \((0,0,0)\) to \((9,9,9)\) in \(\RR^3\) and consider this line integral. (8)
\begin{equation*} \int_\gamma F \cdot ds \end{equation*}For which of the following vector fields is the line integral independent of path and why? When it is independent, what is the potential function?
- \begin{equation*} F(x,y,z) = (2xyz^2, 2x^2z^2, 2x^2yz) \end{equation*}
- \begin{equation*} F(x,y,z) = (2xy^2z, 2x^2yz, x^2y^2z) \end{equation*}
- \begin{equation*} F(x,y,z) = (x^2y^2z, x^2yz^2, x^2y^2z^2) \end{equation*}
- \begin{equation*} F(x,y,z) = (xyz^2, x^2z^2, 2x^2yz) \end{equation*}
- \begin{equation*} F(x,y,z) = (2xyz^2, x^2z^2, 2x^2yz) \end{equation*}
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Let \(\gamma_a\) be a family of paths which are always perpendicular to a vector field \(F\) in \(\RR^3\text{.}\) (6)
Why does the line integral of \(F\) along any \(\gamma_a\) vanish?
If \(F\) is conservative, then \(F = \nabla f\) for some potential \(f\text{.}\) What is the relationship of the potential \(f\) to the paths \(\gamma_a\text{?}\)
If \(F\) is conservative and depends only on the \(y\) coordinate, what can you say about the paths \(\gamma_a\) and the potential \(f\text{?}\)