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Homework 6 Due: Aug. 9

1. Let F~ be some unknown vector field whose curl is given by

.

Compute the flux of ? × F~ over the top half of the unit sphere centered at the origin.

2. Let ~r be the radial vector field ~r = hx,y,zi. The goal of this problem is to show that ~r cannot be written as the curl of some other vector field. In other words, it is impossible that a vector field F~ exists such that ? × F~ = ~r.

(a) Compute the flux of ~r through the unit sphere centered at the origin.

(b) Pretend that there was some vector field F~ such that ? × F~ = ~r. Compute the flux of ? × F~.

(c) Conclude that ~r cannot be written as the curl of another vector field.

3. Find the flux of the vector field

through the surface S bounded by the paraboloic cylinder z = 1 - x2 and the three planes z = 0, y = 0, and z = 2 - y. (Hint: A picture might help!)

1

4. An electrically charged particle sitting at the origin emits a radial electric field E~ given by

where is a constant, and Q is the (constant) charge of the particle.

According to Gauss’ Law from physics, the (electric) flux of E~ through any closed surface S enclosing the particle is given by

The goal of this problem is to prove Gauss’ Law!

(a) Show that ? · E~ = 0.

(b) Let S1 be a sphere centered at the origin whose radius R is small enough so that S1 is completely encased within the unknown surface S. Use the Divergence Theorem to show that

ZZ ZZ

E~ · dS~ = E~ · dS.~

S S1

(Hint: Use the corollary to the Divergence Theorem about the region enclosed by two surfaces.)

(c) Using the unit normal vector

to the sphere S1, compute E~ · ~n.

(d) Compute RRS E~ · dS~ to complete your proof!

5. Consider the unit square R in the uv-plane with vertices (0,0), (1,0), (0,1), (1,1). (a) Show that the change of variables

x = u2 - v2, y = 2uv

takes the four sides of the unit square into a region R1 in the xy-plane bounded by:

1. The line from (0,0) to (1,0).

2. The parabolic arc .

3. The parabolic arc

4. The line from (-1,0) to (0,0). (b) Use this change of variables to calculate

ZZ y dA.

R1

2

Homework 6 Due: Aug. 9

1. Let F~ be some unknown vector field whose curl is given by

.

Compute the flux of ? × F~ over the top half of the unit sphere centered at the origin.

2. Let ~r be the radial vector field ~r = hx,y,zi. The goal of this problem is to show that ~r cannot be written as the curl of some other vector field. In other words, it is impossible that a vector field F~ exists such that ? × F~ = ~r.

(a) Compute the flux of ~r through the unit sphere centered at the origin.

(b) Pretend that there was some vector field F~ such that ? × F~ = ~r. Compute the flux of ? × F~.

(c) Conclude that ~r cannot be written as the curl of another vector field.

3. Find the flux of the vector field

through the surface S bounded by the paraboloic cylinder z = 1 - x2 and the three planes z = 0, y = 0, and z = 2 - y. (Hint: A picture might help!)

1

4. An electrically charged particle sitting at the origin emits a radial electric field E~ given by

where is a constant, and Q is the (constant) charge of the particle.

According to Gauss’ Law from physics, the (electric) flux of E~ through any closed surface S enclosing the particle is given by

The goal of this problem is to prove Gauss’ Law!

(a) Show that ? · E~ = 0.

(b) Let S1 be a sphere centered at the origin whose radius R is small enough so that S1 is completely encased within the unknown surface S. Use the Divergence Theorem to show that

ZZ ZZ

E~ · dS~ = E~ · dS.~

S S1

(Hint: Use the corollary to the Divergence Theorem about the region enclosed by two surfaces.)

(c) Using the unit normal vector

to the sphere S1, compute E~ · ~n.

(d) Compute RRS E~ · dS~ to complete your proof!

5. Consider the unit square R in the uv-plane with vertices (0,0), (1,0), (0,1), (1,1). (a) Show that the change of variables

x = u2 - v2, y = 2uv

takes the four sides of the unit square into a region R1 in the xy-plane bounded by:

1. The line from (0,0) to (1,0).

2. The parabolic arc .

3. The parabolic arc

4. The line from (-1,0) to (0,0). (b) Use this change of variables to calculate

ZZ y dA.

R1

2

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