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Apply the divergence theorem to calculate the flux of the vector field

mathmari

Well-known member
MHB Site Helper
Apr 14, 2013
4,109
Hey!!! :eek:

I have the following exercise:
Apply the divergence theorem to calculate the flux of the vector field $\overrightarrow{F}=(yx-x)\hat{i}+2xyz\hat{j}+y\hat{k}$ at the cube that is bounded by the planes $x= \pm 1, y= \pm 1, z= \pm 1$.

I have done the following...Could you tell me if this is correct?

Flux=$\iint_S{\overrightarrow{F} \cdot \hat{n}} d \sigma=\iiint_D{\nabla \cdot \overrightarrow{F}}dV=\int_{-1}^1 \int_{-1}^1 \int_{-1}^1 {(y-1+2xz)}dxdydz=\int_{-1}^1 \int_{-1}^1{(2y-2)}dydz=\int_{-1}^1{-4}dz=-8$
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,854
Hey!!! :eek:

I have the following exercise:
Apply the divergence theorem to calculate the flux of the vector field $\overrightarrow{F}=(yx-x)\hat{i}+2xyz\hat{j}+y\hat{k}$ at the cube that is bounded by the planes $x= \pm 1, y= \pm 1, z= \pm 1$.

I have done the following...Could you tell me if this is correct?

Flux=$\iint_S{\overrightarrow{F} \cdot \hat{n}} d \sigma=\iiint_D{\nabla \cdot \overrightarrow{F}}dV=\int_{-1}^1 \int_{-1}^1 \int_{-1}^1 {(y-1+2xz)}dxdydz=\int_{-1}^1 \int_{-1}^1{(2y-2)}dydz=\int_{-1}^1{-4}dz=-8$
Yep. Correct. :cool:
 

mathmari

Well-known member
MHB Site Helper
Apr 14, 2013
4,109