MATH 21D

Final | General

Problem 1. Double Integrals

(a) (2 points) Evaluate $\int_0^\pi \int_x^\pi \frac{\sin y}{y} dy \; dx$

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(b) (3 points) A thinp late covers the region $R$ bounded by $y = x$ and $y^2 = x$ in the first quadrant. Assume the plate has constant density $\delta$ and mass $M = 1$. Find $delta$.

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(c) (3 points) Change to polar coordinates but do not evaluate. $\int_0^2 \int_{-\sqrt{1 - (y - 1)^2}}^{0} xy^2 \; dx \; dy$

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Problem 2. Rectangular, Cylindrical and Spherical coordinates

(a) Set-up, but do not evaluate, volume integrals for the following solids.
(i) (2 points) Cylindrical coordinates. $D$ is the right circular cylinder whose base is the circle $r = 3 \cos \theta$ in the $xy$-plane and whose top lies in the plane $z = 5 - x$.

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(ii) (2 points) Spherical coordinates. $D$ lies above the cone $\phi = 3\pi/4$ and between the spheres $x^2 + y^2 + z^2 = 9$ and $x^2 + y^2 + z^2 = 4$.

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(b) (3 points) Verify that for any real number $a > 0$ the rectangular equation $az = \sqrt{x^2 + y^2} $ is equivalent to the spherical equation $\phi = \arctan(a)$.

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Problem 3. Curvature

The curvature $κ$ of a smooth curve $C$ is defined as the magnitude of the derivative of the curve's unit tangent vector $\mathbf{T}$ with respect to arc length $s$.
(a) (2 points) For a smooth parametrized curve $C: \mathbf{r}(t), \quad a \le t \le b$ state the formula for calculating $κ$.

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(b) (4 points) Compute the curvature function $κ = κ(t)$ for the curve $\mathbf{r}(t) = (e^t \cos t)\mathbf{i} + (e^t \sin t) \mathbf{j} + \mathbf{k}$

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Problem 4. Generalizations

(a) (1 points) The line integral $\int_C f(x, y, z) ds$ generalizes the real line integral $\int_a^b f(x) dx$. Consider a paramterized curve $C: \mathbf{r}(t) = < f(t), g(t), h(t) >, \quad a \le t \le b$ Write the equation relating $ds$ and $dt$.

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(b) (3 points) Consider the surface integral $\int \int_S G(x, y, z) d\sigma$ where $S: \mathbf{r}(u, v) = < f(u, v), g(u, v), h(u, v) >, \quad a \le u \le b, c \le v \le d$ Write the surface integral as a paramteric surface integral.

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(c) (4 points) Green's Theorem relates certain line integrals to surface integrals. Use Green's Theorem to write flux and circulation as double integrals in del notation.
Flux across $C =$
Circulation around $C =$

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(d) (4 points) The Divergence Theorem and Stokes' Theorem generalize Green's Theorem as:
Flux of curl =
Flux across $S =$

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Problem 5. The del operator $\nabla$

Let all partial derivatives of the scalar function $f = f(x, y, z)$ be continuous.
(a) (2 points) Write $\nabla$ as a vector.

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(b) (2 points) Write the vector produced by applying the del operator to $f$.

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(c) (2 points) Suppose $f$ is a potential function of a vector field $\mathbf{F}$. Then in terms of $f$
$\mathbf{F} =$ and
$\int_A^B \mathbf{F} \cdot d\mathbf{r} =$

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(d) (4 points) If $\mathbf{F} is as in (c), show $\div \curl \mathbf{F} = | \curl \grad f |$

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Problem 6. Surface Area

Find the surface area of the cone $z = 1 + \sqrt{x^2 + y^2}, z \le 3$

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Problem 7. Exact differential forms

Consider the line integral $\int_A^B df$ where $df$ is the differential form of scalar function $f(x, y, z) = y^2 + xz^2 + 3$ Does the value of the line integral depend on the path from $A$ to $B$? Justify your answer.

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Problem 8. Zero circulation

Show $\mathbf{F} = $ has zero circulation around the boundary of any smooth orientable surface in space.

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