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Calc 3 13.3

 

<== Calculus 3
Number Question Answer
See the work
1. For each of the following vector fields F , decide whether it is conservative or not by computing the appropriate first order partial derivatives. Type in a potential function f (that is, &nabla f = F) with f(0,0) = 0.

A. \mathbf{F} \left( x, y 
ight) = \left( 16 x+ 2 y 
ight) \mathbf{i} +
\left( 2 x+ 10 y
ight) \mathbf{j}
f (x, y) =

B. \mathbf{F} \left( x, y 
ight) = 8 y \mathbf{i} +
9 x \mathbf{j}
f (x, y) =

C. \mathbf{F} \left( x, y 
ight) = \left( 8 \sin y 
ight) \mathbf{i} +
\left( 4 y+ 8  x \cos y 
ight) \mathbf{j}
f (x, y) =

A)

B) Not Conservative

C)
The work
2. Consider the vector field \mathbf{F} \left( x, y, z 
ight) = \left( 3 z+ 4 y
ight)
\mathbf{i} + \left( 2 z+ 4 x 
ight) \mathbf{j} + \left( 2 y+ 3 x 
ight)
\mathbf{k}.

a) Find a function f such that F = &nabla f and f(0,0,0) = 0.
f(x,y,z) =

b) Suppose C is any curve from (0, 0, 0 ) to (1, 1, 1 ). Use part a) to compute the line integral \int_{C} \mathbf{F} \cdot d\mathbf{r}.

A) 3zx +4yx +2zy
B) 9
The work
3. Consider the vector field F (x, y, z) = xi + y j + zk

a) Find a function f such that F = &nabla f and f(0,0,0) = 0.
f(x,y,z) =

b) Use part a) to compute the work done by F on a particle moving along the curve C given by \mathbf{r} \left( t 
ight) = \left( 1+ 2 \sin t
ight) \mathbf{i} + \left( 1+ 4 \sin^2 t 
ight) \mathbf{j} + \left( 1+ \sin^3 t 
ight) \mathbf{k}, \quad 0 \leq t \leq rac{\pi}{2}.

A)
B) 17.5
The work
4. Let \mathbf{F} ( x, y ) = rac{-y \mathbf{i} + x \mathbf{j}}{x^{2} +
y^{2}} and let C be the circle \mathbf{r} \left( t 
ight) = \left( \cos t

ight) \mathbf{i} + \left( \sin t 
ight) \mathbf{j}, 0 &le t &le 2&pi

A. Compute rac{\partial Q}{\partial x}



B. Compute rac{\partial P}{\partial y}



C. Compute \int_C \mathbf{F} \cdot d\mathbf{r}
A)

B)

C) 2&pi
The work
5. Consider the vector field \mathbf{F} = (x^2 + y^2, 5 xy). Compute the line integrals \int_{\mathbf{c}_1} \mathbf{F}\cdot
d\mathbf{r} and \int_{\mathbf{c}_2} \mathbf{F}\cdot d\mathbf{r}, where \mathbf{c}_1(t) = (t, t^2) and \mathbf{c}_2(t) = (t, t) for 0 &le t &le 1.
\int_{\mathbf{c}_1} \mathbf{F}\cdot d\mathbf{r} = =

\int_{\mathbf{c}_2} \mathbf{F}\cdot d\mathbf{r} = =

= 38/15
= 38/15
The work
6. Let \mathbf{F} = (9 xy, 3 y^2) be a vector field in the plane, and C the path y = 2 x^2 joining (0,0) to (1,2) in the plane.

Evaluate \int_C \mathbf{F}\cdot d\mathbf{r}
12.5 The work
7. Suppose that 
abla f(x,y,z) = 2xyze^{x^2}\mathbf{i} + ze^{x^2}\mathbf{j} + ye^{x^2}\mathbf{k}.
If f(0,0,0) = -5, find f(2,2,1).

104.1963 The work
8. question Answer The work
9. question Answer The work