Solving Vector Calculus: (a+2b)∇(∇⋅u) - b∇x∇xu - (3a+2b)c∇T(r)=0

In summary, to solve the given equation for u, first solve for \nabla\times\nabla\times\vec{u} and substitute it into the original equation. Simplify the resulting equation by using the laplacian explicitly and simplifying all individual components. If possible, change the value (3a+2b) to (3a+3b) to further simplify the equation.
  • #1
mcfc
17
0
I'm unsure how to do this problem:

[itex](a + 2b)\nabla(\nabla \cdot \vec u) - b \nabla \times \nabla \times \vec u - (3a + 2b)c\nabla T(r)= \vec 0[/itex]
[itex]\hat u = U_r \hat r + u_\theta \hat \theta +u_z \hat z[/itex]
a,b,c constants
how would I solve this for u?
 
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  • #2
i know that [tex]\nabla[/tex]([tex]\nabla\bullet\vec{u}[/tex]) - [tex]\nabla\times\nabla\times\vec{u}[/tex] = [tex]\nabla^{2}\vec{u}[/tex]

i dunno, what about this?..

Solve this equation for [tex]\nabla\times\nabla\times\vec{u}[/tex], then substitute into your equation, and simplify the resulting equation. The first term of your equation should reduce to (a+b)[tex]\nabla[/tex]([tex]\nabla\bullet\vec{u}[/tex]) +b[tex]\nabla^{2}\vec{u}[/tex] - (3a+2b)c[tex]\nabla[/tex]T(r) = 0.

You should be able to write out the laplacian explicitly and simplify all the individual components. If you can find some way to turn the value (3a+2b) into (3a+3b) you could probably simplify things a lot.
 

Related to Solving Vector Calculus: (a+2b)∇(∇⋅u) - b∇x∇xu - (3a+2b)c∇T(r)=0

1. What is vector calculus?

Vector calculus is a branch of mathematics that deals with vector fields, which are mathematical objects that have both magnitude and direction. It involves the study of operations such as differentiation and integration on vector fields.

2. What does the expression (a+2b)∇(∇⋅u) - b∇x∇xu - (3a+2b)c∇T(r)=0 mean?

This expression is a vector calculus equation that represents the equilibrium of forces in a given system. The first term, (a+2b)∇(∇⋅u), represents the divergence of the gradient of a vector field u. The second term, b∇x∇xu, represents the curl of the curl of u. The third term, (3a+2b)c∇T(r), represents the gradient of the temperature field T multiplied by a constant vector c. The overall equation states that the sum of these three terms is equal to zero.

3. How is vector calculus used in science?

Vector calculus is used in various scientific fields, including physics, engineering, and computer graphics. It is often used to describe and analyze physical phenomena that involve quantities with both magnitude and direction, such as force, velocity, and electric and magnetic fields.

4. What are some real-world applications of vector calculus?

Some real-world applications of vector calculus include predicting the motion of objects in fluid dynamics, modeling electromagnetic fields in electrical engineering, and creating 3D computer graphics in computer science. It is also used in fields such as meteorology, geology, and economics.

5. Are there any tools or software that can assist with solving vector calculus problems?

Yes, there are various tools and software available that can assist with solving vector calculus problems. Some popular options include WolframAlpha, MATLAB, and Maple. These tools can help with computations, graphing, and visualizing vector fields, making it easier to solve complex vector calculus equations.

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