Vector Calculus   Sample Final Examination #1

Warning to Instructors: Question 2 may involve more linear algebra than you are assuming, so modify it accordingly (eg, by deleting or changing parts (b) and (c).

1.

Let $f (x, y) = e ^{ x y} \sin (x + y)$.

(a)

In what direction, starting at $(0, \pi / 2)$, is f changing the fastest?

(b)

In what directions starting at $(0, \pi / 2)$ is f changing at 50% of its maximum rate?

(c)

Let ${\bf c} (t)$ be a flow line of ${\bf F} = \nabla f$ with ${\bf c} (0) = (0, \pi/2 )$. Calculate

$$\left. \frac{d }{d t} [ f (c (t)) ] \right\vert _{ t = 0} .$$

2.

Let $f : { \mathbb R}^3 \rightarrow { \mathbb R}^3$ be a given mapping and write f (x, y, z) = (u (x, y, z) , v (x, y, z), w (x, y, z)). Let $g : { \mathbb R}^3 \rightarrow { \mathbb R}^3$ be defined by g (u,v,w) = (u - v, u + w, w + v ) and let $h = g \circ f$.

(a)

Write a formula for the derivative matrix ${\bf D} h$.

(b)

Show that ${\bf D} h$ cannot have rank 3 at any point (x, y, z).

(c)

Show that ${\bf D} h$ has an eigenvalue zero at every (x, y, z).

3.

Extremize f (x, y, z) = x subject to the constraints

$$x^2 + y^2 + z^2 = 1 \quad \mbox{and} \quad x + y + z = 1.$$

4.

(a)

Evaluate

$$\int \!\!\! \int \!\!\! \int _D \exp [(x^2 + y^2 + z^2) ^{ 3/2}] \, d x\, d y\, d z$$

where D is the region defined by $1 \leq x^2 + y^2 + z^2 \leq 2$and $z \geq 0$.

(b)

Sketch or describe the region of integration for

$$\int^1_0 \int^x_0 \int^y_0 f (x, y, z) d z\, d y\, d x,$$

and interchange the order to $dy\, dx \, dz$.

5.

Let ${\bf G} (x, y) = (x e ^{ x^2 + y^2} + 2 xy) {\bf i} + (ye ^{ x^2 + y^2} + x^2) {\bf j}$.

(a)

Show that ${\bf G} = {\bf \nabla} f$ for some f; find such an f.

(b)

Use (a) to show that the line integral of ${\bf G}$ around the edge of the triangle with vertices (0,0) , (0,1), (1,0) is zero.

(c)

State Green's theorem for the triangle in (b) and a vector field ${\bf F}$ and verify it for the vector field ${\bf G}$ above.

6.

Let W be the three dimensional region under the graph of $f (x, y) = \exp (x^2 + y^2)$and over the region in the plane defined by $1 \leq x^2 + y^2 \leq 2$.

(a)

Find the volume of W.

(b)

Find the flux of the vector field ${\bf F} = (2 x - x y) {\bf i} - y {\bf j} + yz {\bf k}$ out of the region W.

7.

Let C be the curve x² + y² = 1 lying in the plane z = 1. Let ${\bf F} = (z - y) {\bf i} + y {\bf k}$.

(a)

Calculate $\nabla \times {\bf F}$.

(b)

Calculate ${\displaystyle \int _C {\bf F} \cdot d {\bf s} }$ using a parametrization of C and a chosen orientation for C.

(c)

Write $C = \partial S$ for a suitably chosen surface S and, applying Stokes' theorem, verify your answer in (b) .

(d)

Consider the sphere with radius $\sqrt{2}$ and center the origin. Let $S ^\prime$ be the part of the sphere that is above the curve (i.e., lies in the region $z \geq 1$), and has C as boundary. Evaluate the surface integral of $\nabla \times {\bf F}$over $S ^\prime$. Specify the orientation you are using for $S ^\prime$.