# MATH 251: Calculus 3, SET8

## 16: Vector Calculus

### 16.5: Curl and Divergence

These problems are done with the CAS. See Hand Solutions for details.

#### 1. [1109/2]

F = [0 x^3 * y * z^2 y^4 * z^3]

F = curl_F = curl(F, [x y z])' % MATLAB's curl command returns a column vector, so we transpose.

curl_F = div_F = divergence(F, [x y z])

div_F = #### 2. [1109/4]

F = [sin(y*z) sin(x*z) sin(x*y)]

F = curl_F = curl(F, [x y z])'

curl_F = div_F = divergence(F, [x y z])

#### 3. [1109/6]

F = [log(2*y + 3*z) log(x + 3*z) log(x + 2*y)]

F = curl_F = curl(F, [x y z])'

curl_F =

div_F = divergence(F, [x y z])

#### 4. [1109/8]

F = [atan(x*y) atan(y*z) atan(x*z)]

F = curl_F = curl(F, [x y z])'

curl_F =

div_F = divergence(F, [x y z])

div_F =

#### 5. [1109/10]

F = [x y 0] % This is the vector field depicted in the figure.

F = curl_F = curl(F, [x y z])'

curl_F = div_F = divergence(F, [x y z])

% (b) and curl(F) = [0 0 0], the zero vector.

#### 6. [1109/14]

F = [x*y*z^4 x^2*z^4 4*x^2*y*z^3]

F = curl_F = curl(F, [x y z])' % Since curl(F) is not the zero vector on R^3, F is not conservative.

curl_F = f = potential(F, [x y z]) % Indeed, F has no potential function.

#### 7. [1109/16]

F = [1 sin(z) y*cos(z)]

F = curl_F = curl(F, [x y z])' % Since curl(F) is the zero vector on R^3, F is conservative.

curl_F = f = potential(F, [x y z]) % Here is a potential function f for vector field F.

f = #### 8. [1109/18]

F = exp(x)*[sin(y*z) z*cos(y*z) y*cos(y*z)]

F = curl_F = curl(F, [x y z])' % Since curl(F) is the zero vector on R^3, F is conservative.

curl_F = f = potential(F, [x y z]) % Here is a potential function f for vector field F.

f = #### 9. [1109/20]

Let F. Assume there exists a vector field G on such that curl(G) = F. Then by Theorem 11 on page 1106 of the textbook, div(curl(G)) = div(F) = 0. But div(F) = 3 (see below), a contradiction. Hence there is no such vector field G. F = [x y z]

F = div_F = divergence(F, [x y z])

#### 10. [1109/22]

syms x y z f(y,z) g(x,z) h(x,y)

F(x,y,z) = [f(y,z) g(x,z) h(x,y)]

F(x, y, z) = div_F = divergence(F, [x y z])

% Since div(F) = 0, F is said to be incompressible (page 1107 of textbook).