Let $\vec{F}=\operatorname{grad} f$ , where $f ( x , y ) = \ln \left( 2 x ^ { 2 } + 3 y ^ { 2 } + 1 \right)$ (a)Evaluate the line integral $\int _ { C } \vec { F } \cdot \overline { d r }$ , where C is the line from (0, 0)to (2, -4).
(b)Do you expect the line integral of $\vec{F}$ along the parabola y = x²-4x, 0 $\le$ x $\le$ 2 to equal to the answer to (a)?

Question

Let

\vec{F}=\operatorname{grad} f

, where

f ( x , y ) = \ln \left( 2 x ^ { 2 } + 3 y ^ { 2 } + 1 \right)

(a)Evaluate the line integral

\int _ { C } \vec { F } \cdot \overline { d r }

, where C is the line from (0, 0)to (2, -4).
(b)Do you expect the line integral of

\vec{F}

along the parabola y = x²-4x, 0

\le

x

\le

2 to equal to the answer to (a)?

Quizplus · Accepted Answer

The Answer of Let $\vec{F}=\operatorname{grad} f$ , where $f ( x , y ) = \ln \left( 2 x ^ { 2 } + 3 y ^ { 2 } + 1 \right)$ (a)Evaluate the line integral $\int _ { C } \vec { F } \cdot \overline { d r }$ , where C is the line from (0, 0)to (2, -4). (b)Do you expect the line integral of $\vec{F}$ along the parabola y = x²-4x, 0 $\le$ x $\le$ 2 to equal to the answer to (a)?

Let F⃗=grad⁡f\vec{F}=\operatorname{grad} fF=gradf , Where f(x,y)=ln⁡(2x2+3y2+1)f ( x , y ) = \ln \left( 2 x ^ { 2 } + 3 y ^ { 2 } + 1 \right)f(x,y)=ln(2x2+3y2+1)

Let $\vec{F}=\operatorname{grad} f$ , Where $f ( x , y ) = \ln \left( 2 x ^ { 2 } + 3 y ^ { 2 } + 1 \right)$