5. Vector Calculus

Fundamental Theorem For Line Integrals — Quiz

Test your understanding of fundamental theorem for line integrals with 5 practice questions.

Practice Questions

Question 1

Suppose the vector field $\mathbf{F}(x,y,z) = (2x, 2y, 2z)$. Which of the following is the correct potential function $f(x,y,z)$ such that $\mathbf{F} = \nabla f$?

Question 2

Which condition must a vector field $\mathbf{F}(x,y,z)$ satisfy to ensure that the line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$ is path independent?

Question 3

Evaluate the line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$ for $\mathbf{F}(x,y,z) = (3x^2, 4y^3, 5z^4)$ along the curve $C$ from $(1,1,1)$ to $(2,2,2)$ if $\mathbf{F}$ is conservative.

Question 4

Let $\mathbf{F}(x,y) = (y, -x)$. Is $\mathbf{F}$ a conservative vector field?

Question 5

Which of the following is true for a conservative vector field $\mathbf{F}(x,y,z)$?