6. Cauchy’s Theorem

Consequences Of Path Independence — Quiz

Test your understanding of consequences of path independence with 5 practice questions.

Practice Questions

Question 1

If $\int_\gamma f(z)\,dz$ is path independent on a domain $D$, what is $\oint_C f(z)\,dz$ for any closed curve $C$ in $D$?

Question 2

Suppose path independence lets us define $F(z)=\int_{z_0}^z f(w)\,dw$. What is $F'(z)$?

Question 3

Let $\gamma_1$ and $\gamma_2$ be two paths in $D$ from $a$ to $b$. If the integral of $f$ is path independent, what is true?

Question 4

If a line integral of $f$ is path independent, what determines the value of the integral from $a$ to $b$?

Question 5

If $\gamma^{-1}$ is the path $\gamma$ traced in the reverse direction, how does $\int_{\gamma^{-1}} f(z)\,dz$ compare with $\int_\gamma f(z)\,dz$?