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Section 11.3 Assignment 3

  1. Explain how vector fields might be useful in each of the following situations. (6)

    1. Planning a wind farm.

    2. Shielding a solenoid.

    3. Tracking shoals of fish.

  2. Draw these vector fields. (You can check this with a computer, but I want to see some of the calculations behind your drawing.)

    1. \begin{equation*} F(x,y) = (x^2 - y, y^2 -x) \end{equation*}

    2. \begin{equation*} F(x,y) = \left( \frac{1}{x+y}, \frac{1}{x-y} \right) \end{equation*}

    3. \begin{equation*} F(x,y) = (x \cos (\pi y), y \cos (\pi x)) \end{equation*}

  3. Calculate the integral curves for the following vector fields. (8)

    1. \begin{equation*} F(x,y) = \left( \frac{x}{2}, \frac{y}{4} \right) \end{equation*}

    2. \begin{equation*} F(x,y) = \left( \frac{1}{x^2}, \frac{1}{y^3} \right) \end{equation*}

  4. What do integral curves mean for the following vector fields? (9)

    1. The vector field of ocean currents. (2D or 3D)

    2. The vector field of gravitational force per unit mass around a gravitational source. (3D)

    3. The vector field of temperature gradients over a landscape. (2D)

  5. 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)

    1. \begin{equation*} \nabla \times (\nabla f) = (0,0,0) \end{equation*}

    2. \begin{equation*} \nabla \cdot (\nabla \times F) = 0 \end{equation*}

  6. 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)

  7. Calculate the line integrals.

    \begin{equation*} \int_\gamma F \cdot ds \end{equation*}

    for these \(F\) and \(\gamma\text{.}\) (16)

    1. \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*}

    2. \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*}

    3. \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*}

    4. \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*}

  8. 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?

    1. \begin{equation*} F(x,y,z) = (2xyz^2, 2x^2z^2, 2x^2yz) \end{equation*}

    2. \begin{equation*} F(x,y,z) = (2xy^2z, 2x^2yz, x^2y^2z) \end{equation*}

    3. \begin{equation*} F(x,y,z) = (x^2y^2z, x^2yz^2, x^2y^2z^2) \end{equation*}

    4. \begin{equation*} F(x,y,z) = (xyz^2, x^2z^2, 2x^2yz) \end{equation*}

    5. \begin{equation*} F(x,y,z) = (2xyz^2, x^2z^2, 2x^2yz) \end{equation*}

  9. Let \(\gamma_a\) be a family of paths which are always perpendicular to a vector field \(F\) in \(\RR^3\text{.}\) (6)

    1. Why does the line integral of \(F\) along any \(\gamma_a\) vanish?

    2. If \(F\) is conservative, then \(F = \nabla f\) for some potential \(f\text{.}\) What is the relationship of the potential \(f\) to the paths \(\gamma_a\text{?}\)

    3. If \(F\) is conservative and depends only on the \(y\) coordinate, what can you say about the paths \(\gamma_a\) and the potential \(f\text{?}\)