Question

The figure shows a curve \(C\)and contour map of a function \(f\)whose gradient is continuous. Find\(\int_C \nabla {\rm{f}} \cdot {\rm{dr}}\).

Short answer

The value of\(\int_C \nabla {\rm{f}} \cdot {\rm{dr = 40}}\).

Step-by-step solution

Line integrals fundamental theorem.

LINE INTEGRALS FUNDAMENTAL THEOREM

Let \(C\)be a smooth curve defined by the vector function. It \(f\)is a differentiable function \(C\)with a continuous gradient vector\(\nabla {\rm{f}}\), then

\(\int_{\rm{C}} \nabla {\rm{f}} \cdot {\rm{dr = f(r(b)) = f(r(a))}}\)

To find\(\int_C \nabla  {\rm{f}} \cdot {\rm{dr}}\).

Can you see from the figure in the text, \(C\)is a smooth curve, Because \(\nabla {\rm{f}}\)is continuous in the problem formulation, \(f\)is differentiable, and the criteria of the fundamental theorem for line integrals hold.

As a result, \(\int_C \nabla f \cdot d{\bf{r}}\)equals the difference between the values of \(f\)at the path's final and beginning locations.

\(\begin{array}{c}\int_C \nabla {\rm{f}} \cdot {\rm{dr = 50 - 10}}\\{\rm{ = 40}}\end{array}\)

Therefore, the value of\(\int_C \nabla {\rm{f}} \cdot {\rm{dr = 40}}\).