Let $\mathbf { F } ( x , y , z ) = \frac { x \mathbf { i } + y \mathbf { j } + z \mathbf { k } } { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 3 / 2 } }$ . Evaluate the line integral $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ , where C is the curve of intersection of the paraboloid $x ^ { 2 } + y ^ { 2 } = 2 z$ and the cylinder $x ^ { 2 } + y ^ { 2 } = 2 x$ .
A) $1$

B)$- 1$
C)$\sqrt { 2 }$
D)$- \sqrt { 2 }$
E)$\sqrt { 3 }$
F)$- \sqrt { 3 }$

G)$2$

H)$0$

Question

Let $\mathbf { F } ( x , y , z ) = \frac { x \mathbf { i } + y \mathbf { j } + z \mathbf { k } } { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 3 / 2 } }$ . Evaluate the line integral $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ , where C is the curve of intersection of the paraboloid $x ^ { 2 } + y ^ { 2 } = 2 z$ and the cylinder $x ^ { 2 } + y ^ { 2 } = 2 x$ .
A) $1$

B)$- 1$
C)$\sqrt { 2 }$ 
D)$- \sqrt { 2 }$ 
E)$\sqrt { 3 }$ 
F)$- \sqrt { 3 }$

G)$2$

H)$0$

Quizplus · Accepted Answer

The vector field $\mathbf { F } ( x , y , z ) = \frac { x \mathbf { i } + y \mathbf { j } + z \mathbf { k } } { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 3 / 2 } }$ is the gradient of the scalar function $\phi(x, y, z) = \frac{1}{\sqrt{x^2 + y^2 + z^2}}$, which means $\mathbf{F} = \nabla \phi$. Since $\mathbf{F}$ is a conservative vector field (the gradient of a scalar field), the line integral of $\mathbf{F}$ over any closed curve $C$ is zero. The curve described is closed because it is the intersection of a paraboloid and a cylinder, forming a loop. Therefore, the line integral $\int _ { C } \mathbf { F } \cdot d \mathbf { r } = 0$.

Let Evaluate the Line Integral ∫CF⋅dr\int _ { C } \mathbf { F } \cdot d \mathbf { r }∫C​F⋅dr

Let Evaluate the Line Integral $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$