Circulation of a 3d vector field

In summary, the conversation discusses a vector field with differentiable functions h and g, and a closed curve C in a horizontal plane. The question asks for a proof that the circulation of the vector field around C is dependent only on the enclosed area and the value of h at a specific point. The suggested solution involves parameterizing the curve and using a line integral to factor out constants and recognize a formula for area.
  • #1
cummings12332
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Homework Statement


consider the vector field v(x,y,z)=(-h(z)y,h(z)x,g(z)) wherer h:R->R and g:R—>R are differentiable .Let C be a closed curve in the horizontal plane z=z0.show that the circulation of v around C depends only on the area of the reion enclosed by C in the given plane and h(Z0)


The Attempt at a Solution


i know it is simple,but i don't know how to star it. should i parameterize the curve c ? but how? could someone give me some details hints?
 
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  • #2
cummings12332 said:

Homework Statement


consider the vector field v(x,y,z)=(-h(z)y,h(z)x,g(z)) wherer h:R->R and g:R—>R are differentiable .Let C be a closed curve in the horizontal plane z=z0.show that the circulation of v around C depends only on the area of the reion enclosed by C in the given plane and h(Z0)


The Attempt at a Solution


i know it is simple,but i don't know how to star it. should i parameterize the curve c ? but how? could someone give me some details hints?

Parameterize any way you like, say [x(t),y(t),z0], for t in [0,1]. Write down the line integral that defines "circulation" around your curve. Factor out constants. Do you recognize a formula for area in what's left?
 

1. What is a 3d vector field?

A 3d vector field is a mathematical concept used to describe the behavior of a physical quantity, such as velocity or force, in three-dimensional space. It is represented by arrows or vectors, with each vector representing the magnitude and direction of the quantity at a specific point in space.

2. How is circulation defined in a 3d vector field?

Circulation in a 3d vector field is a measure of the flow of the vector field along a closed path. It is calculated by taking the dot product of the vector field with the tangent vector of the path at each point, and then integrating over the entire path.

3. What is the significance of circulation in fluid dynamics?

In fluid dynamics, circulation is used to understand the movement of fluids and the forces acting on them. It can help determine the strength and direction of vortices, as well as the lift and drag forces on objects in a fluid.

4. How is circulation related to the concept of a conservative vector field?

In a conservative vector field, the circulation around any closed path is always zero. This means that the work done by the vector field on a particle moving along a closed path is independent of the path taken, and only depends on the starting and ending points. In other words, the path does not affect the total energy of the system.

5. What are some real-life applications of studying circulation in a 3d vector field?

Studying circulation in a 3d vector field has many practical applications, such as understanding the flow of air around airplanes and optimizing their design, predicting the movement of ocean currents and their impact on marine life, and analyzing the flow of blood in the human body to diagnose and treat cardiovascular diseases.

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