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

In summary, the conversation discusses the application of the divergence theorem to calculate the flux of a vector field on a cube bounded by certain planes. The calculation is shown step by step and it is confirmed to be correct.
  • #1
mathmari
Gold Member
MHB
5,049
7
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=$\iint_S{\overrightarrow{F} \cdot \hat{n}} d \sigma=\iiint_D{\nabla \cdot \overrightarrow{F}}dV=\int_{-1}^1 \int_{-1}^1 \int_{-1}^1 {(y-1+2xz)}dxdydz=\int_{-1}^1 \int_{-1}^1{(2y-2)}dydz=\int_{-1}^1{-4}dz=-8$
 
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  • #2
mathmari said:
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:
 
  • #3
I like Serena said:
Yep. Correct. :cool:

Great! Thanks a lot! :eek:
 

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

1. What is the divergence theorem?

The divergence theorem, also known as Gauss's theorem, is a fundamental theorem in vector calculus that relates the flux of a vector field through a closed surface to the divergence of the vector field within the volume enclosed by the surface.

2. How do you apply the divergence theorem to calculate flux?

To apply the divergence theorem, you need to first calculate the divergence of the given vector field. Then, you need to find the volume enclosed by the closed surface and integrate the divergence over this volume. The result of this integration will give you the flux of the vector field through the surface.

3. What is the significance of calculating flux using the divergence theorem?

The divergence theorem provides a more efficient method for calculating flux as it reduces the calculation to a single integral instead of evaluating the flux at each point on the surface. It also helps in understanding the flow of a vector field through a given surface and its relationship to the divergence of the field.

4. Can the divergence theorem be applied to any vector field?

Yes, the divergence theorem can be applied to any vector field as long as the field is continuous and differentiable within the volume enclosed by the closed surface.

5. What are some real-world applications of the divergence theorem?

The divergence theorem has various applications in physics and engineering, such as calculating the flux of electric or magnetic fields, analyzing fluid flow through a closed surface, and understanding the flow of heat in a system. It is also used in computer graphics and image processing to simulate fluid flow or heat transfer in a given volume.

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