1. Let P = P(x, y, z), Q = Q(x, y, z), and R = R(x, y, z) be functions of three variables defined on a subset E of space. A vector

field over E is defined as the function F(x, y, z) = P(x, y, z) + Q(x, y, z) + R(x, y, z).

Determine whether the provided statement is true or false.

2. The gradient of a function f is an example of a vector field.

Determine whether the provided statement is true or false.

3. Consider the vector field F(x, y) = xi + 2yj.

Which of the following vector fields correctly graphs F(x, y)?

Choose the correct vector field.

4. Consider the vector field F = F(x, y) = yi-j.

Which of the following vector fields correctly graphs F(x, y)?

Choose the correct vector field.

5. Consider the vector field F = F(x, y) = 2i.

Which of the following vector fields correctly graphs F(x, y)?

Choose the correct vector field.

6. Consider the vector field F = F(x, y) = -4i + 3j.

Which of the following vector fields correctly graphs F(x, y)?

Choose the correct vector field.

7. Consider the vector field F = F(x, y, z) = zk.

Which of the following vector fields correctly graphs F(x, y, z)?

Choose the correct vector field.

8. Consider the vector field F = F(x, y, z) = xi.

Which of the following vector fields correctly graphs F(x, y, z)?

Choose the correct vector field.

9. Find the gradient vector field of function f.

f(x,y) = xe^13y

(Give your answer using component form or standard basis vectors. Express numbers in exact form. Use symbolic notation and

fractions where needed.)

10. Find the gradient vector field of function ƒ.

f(x, y, z) = x^7y + 4xyz³

(Give your answer using component form or standard basis vectors. Express numbers in exact form. Use symbolic notation and

fractions where needed.)

11. (a) Which of the following correctly describes the vector field F(x, y) = -yi + xj?

F(x, y) is a set of vectors

tangent to circles centered at the origin.

O parallel to the y-axis.

O parallel to the x-axis.

O parallel to radius vectors of circles centered at the origin.

(b) Which of the following correctly describes the magnitude of a vector in the field F(x, y) = -yi + xj?

The magnitude of each vector in the field

O is proportional to the distance of the vector from the x-axis.

O equals the radius squared of the circle.

O equals the radius of the circle.

O is proportional to the distance of the vector from the y-axis.

12. Consider a very small object of mass m kg. The distance from this object is denoted by r and measured in meters (m). The

gravitational potential due to the mass m is the scalar function u = - Gr, where G = 6.67 x 10-¹¹ N m²/kg² and

r = xi + yj + zk. If the gravitational field g = -Vu, then what does g equal in terms of G, m, r, and r?