Flux of Vector Field Across Surface

In summary, the conversation discusses the computation of a flux of a vector field (grad x F) across a surface obtained by gluing a cylinder and hemispherical cap together. The gradient of F is found to be <z + 2xy, x^3z +1, 2zx^4> and the individual volumes are calculated using cylindrical and spherical coordinates. The integration of grad F over the surface is necessary and can be done separately for the cylinder and hemisphere.
  • #1
cheeee
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Homework Statement



Compute the flux of vector field (grad x F) where F = (xz+x^2y + z, x^3yz + y, x^4z^2)
across the surface S obtained by gluing the cylinder z^2 + y^2 = 1 (x is > or eq to 0 and < or eq to 1) with the hemispherical cap z^2 + y^2 (x-1)^2 = 1 (x > or eq to 1) oriented in such a way that the unit normal at (1,0,0) is given by i.

My approach to find the flux is to take the triple integral of the gradient of F * dV. I found the gradient to be <z + 2xy, x^3z +1, 2zx^4> and I set up the integral but I am a bit confused as to what the limits should be. Should I be even taking a triple integral??
 
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  • #2
also... can I use cylindrical coordinates for the volume of the cylinder then use spherical coordinates for the volume of the cap and add them together?
 
  • #3
How are you taking the triple integral? The "gradient of F" is a vector quantity and dV is not. I thought perhaps you were using the divergence theorem but you say nothing about the divergence of grad F (which would be [itex]\nabla^2 F[/itex]).

If you are doing this directly (not using the divergence theorem), you need to integrate grad F over the surface of the figure. Yes, you can do the cylinder and hemisphere separately.
 

1. What is the definition of flux of a vector field across a surface?

The flux of a vector field across a surface is a measure of the flow of the vector field through the surface. It is a scalar quantity that represents the amount of fluid, energy, or other physical quantity passing through a given surface in a given amount of time.

2. How is the flux of a vector field calculated?

The flux of a vector field is calculated by taking the dot product of the vector field and the unit normal vector to the surface, and then integrating this product over the surface.

3. What is the significance of the direction of the flux of a vector field?

The direction of the flux of a vector field is important because it indicates the direction in which the fluid or energy is flowing through the surface. If the flux is positive, it means the fluid or energy is flowing out of the surface, while a negative flux indicates flow into the surface.

4. How does the shape and orientation of a surface affect the flux of a vector field?

The shape and orientation of a surface can have a significant impact on the flux of a vector field. For example, a surface that is perpendicular to the direction of the vector field will have a higher flux than a surface that is parallel to the vector field. Additionally, a curved surface may have varying flux values at different points along its surface.

5. Can the flux of a vector field be negative?

Yes, the flux of a vector field can be negative, which indicates flow into the surface. This can occur if the vector field is directed towards the surface, or if the surface is oriented in a way that causes the dot product of the vector field and the unit normal vector to be negative.

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