Suppose That $f ( x , y , z )$ Is Defined and Differentiable Everywhere and Satisfies the Differential Equation

Short Answer

Suppose that $f ( x , y , z )$ is defined and differentiable everywhere and satisfies the differential equation $x \frac { \partial f } { \partial x } + y \frac { \partial f } { \partial y } + z \frac { \partial f } { \partial z } = 0$ .Let ${ \vec { F } } = f ( x , y , z ) \vec { r }$ , where $\vec { r } = x \vec { i } + y \vec { j } + z \vec { k }$ .Suppose that S is a closed surface and W is its interior.Find Q in the following equation: $\int _ { S } \vec { F } \cdot d \vec { A } = Q \int _ { W } f ( x , y , z ) d V$ .