State the Fundamental Theorem of Calculus for Line Integrals.
A)Suppose C is a piece-wise smooth oriented path with starting point P and end point Q.Then $\int _ { C } \operatorname { grad } f \cdot d \vec { r } = f ( Q ) - f ( P )$
B)Suppose C is a oriented path with starting point P and end point Q.Then $\int _ { C } \operatorname { grad } f \cdot d \vec { r } = f ( Q ) - f ( P )$
C)Suppose C is a piece-wise smooth oriented path with starting point P and end point Q.If f is a function whose gradient is continuous on the path C, then $\int _ { C } \operatorname { grad } f \cdot d \vec { r } = f ( Q ) - f ( P )$
D)Suppose C is a piece-wise smooth oriented path with starting point P and end point Q.If f is a function whose gradient is continuous on the path C, then $\int _ { C } \operatorname { grad } f \cdot d \vec { r } = f ( P ) - f ( Q ) .$

Question

Quizplus · Accepted Answer

The Fundamental Theorem of Calculus for Line Integrals states that if $C$ is a piece-wise smooth oriented path from point $P$ to point $Q$ and $f$ is a function whose gradient is continuous on $C$, then the line integral of the gradient of $f$ along $C$ equals $f(Q) - f(P)$. This captures the essence of evaluating the function at the endpoints of the path.

State the Fundamental Theorem of Calculus for Line Integrals ∫Cgrad⁡f⋅dr⃗=f(Q)−f(P)\int _ { C } \operatorname { grad } f \cdot d \vec { r } = f ( Q ) - f ( P )∫C​gradf⋅dr=f(Q)−f(P)

State the Fundamental Theorem of Calculus for Line Integrals $\int _ { C } \operatorname { grad } f \cdot d \vec { r } = f ( Q ) - f ( P )$