Gilbert Strang; Edwin “Jed” Herman

Review Exercises

True or False? Justify your answer with a proof or a counterexample.

Vector field $F (x, y) = x^{2} y i + y^{2} x j$ is conservative.

428.

For vector field $F (x, y) = P (x, y) i + Q (x, y) j,$ if $P_{y} (x, y) = Q_{x} (x, y)$ in open region $D,$ then $\int_{\partial D} P d x + Q d y = 0 .$

429.

The divergence of a vector field is a vector field.

430.

If $curl F = 0,$ then $F$ is a conservative vector field.

Draw the following vector fields.

431.

$F (x, y) = \frac{1}{2} i + 2 x j$

432.

$F (x, y) = \sqrt{\frac{y i + 3 x j}{x^{2} + y^{2}}}$

Are the following the vector fields conservative? If so, find the potential function $f$ such that $F = \nabla f .$

433.

$F (x, y) = y i + (x - 2 e^{y}) j$

434.

$F (x, y) = (6 x y) i + (3 x^{2} - y e^{y}) j$

435.

$F (x, y, z) = (2 x y + z^{2}) i + (x^{2} + 2 y z) j + (2 x z + y^{2}) k$

436.

$F (x, y, z) = (e^{x} y) i + (e^{x} + z) j + (e^{x} + y^{2}) k$

Evaluate the following integrals.

437.

$\int_{C} x^{2} d y + (2 x - 3 x y) d x,$ along $C : y = \frac{1}{2} x$ from (0, 0) to (4, 2)

438.

$\int_{C} y d x + x y^{2} d y,$ where $C : x = \sqrt{t}, y = t - 1, 0 \leq t \leq 1$

439.

$\iint_{S} x y^{2} d S,$ where S is surface $z = x^{2} - y, 0 \leq x \leq 1, 0 \leq y \leq 4$

Find the divergence and curl for the following vector fields.

440.

$F (x, y, z) = 3 x y z i + x y e^{z} j - 3 x y k$

441.

$F (x, y, z) = e^{x} i + e^{x y} j + e^{x y z} k$

Use Green’s theorem to evaluate the following integrals.

442.

$\int_{C} 3 x y d x + 2 x y^{2} d y,$ where C is a square with vertices (0, 0), (0, 2), (2, 2) and (2, 0) oriented counterclockwise.

443.

$\int_{C} 3 y d x + (x + e^{y}) d y,$ where C is a circle centered at the origin with radius 3

Use Stokes’ theorem to evaluate $\int \int_{S} curl F \cdot d S .$

444.

$F (x, y, z) = y i - x j + z k,$ where $S$ is the upper half of the unit sphere

445.

$F (x, y, z) = y i + x y z j - 2 z x k,$ where $S$ is the upward-facing paraboloid $z = x^{2} + y^{2}$ lying in cylinder $x^{2} + y^{2} = 1$

Use the divergence theorem to evaluate $\int \int_{S} F \cdot d S .$

446.

$F (x, y, z) = (x^{3} y) i + (3 y - e^{x}) j + (z + x) k,$ over cube $S$ defined by $−1 \leq x \leq 1,$ $0 \leq y \leq 2,$ $0 \leq z \leq 2$

447.

$F (x, y, z) = (2 x y) i + (− y^{2}) j + (2 z^{3}) k,$ where $S$ is bounded by paraboloid $z = x^{2} + y^{2}$ and plane $z = 2$

448.

Find the amount of work performed by a 50-kg woman ascending a helical staircase with radius 2 m and height 100 m. The woman completes five revolutions during the climb.

449.

Find the total mass of a thin wire in the shape of an upper semicircle with radius $\sqrt{2,}$ and a density function of $ρ (x, y) = y + x^{2} .$

450.

Find the total mass of a thin sheet in the shape of a hemisphere with radius 2 for $z \geq 0$ with a density function $ρ (x, y, z) = x + y + z .$

451.

Use the divergence theorem to compute the value of the flux integral over the unit sphere with $F (x, y, z) = 3 z i + 2 y j + 2 x k .$