Vector and Tensor problems

[slashragnarok]

Hi guys and gals. I'm stuck on the following problems. Please help me out.

1. Show that curl v is twice the local angular velocity (w), where v is the velocity vector of the fluid.

2. Prove that I:v= div v where I is a unit tensor.

3. Explain why alternating unit tensor is very important in order to describe the cross product of two vectors.

4. If div E=0, div H=0, curl E=- $$\partial$$H/$$\partial$$ t , curl H= $$\partial$$E /$$\partial$$t then show that E and H satisfy $$\nabla$$²u= $$\partial$$²u/$$\partial$$t²

 

[pat_connell]

where u is the scalar potential.Answer 1: Curl v can be written as curl v = (v · ∇)v – ∇(v · v). Since angular velocity is defined as w= v · ∇v, then we can rewrite curl v as curl v = 2w - ∇(v · v). The only term that remains is 2w, thus curl v is twice the local angular velocity (w). Answer 2: Let v be a vector field with components v1, v2 and v3, and I be a unit tensor. Then, I:v = I11v1 + I22v2 + I33v3. Using the properties of the unit tensor, we can simplify this expression to I:v = v1 + v2 + v3. This is exactly the same as the divergence of the vector field v, i.e. div v = v1 + v2 + v3. Therefore, I:v = div v. Answer 3: The alternating unit tensor is an important tool in describing the cross product of two vectors. When two vectors are multiplied together, the result is a third vector that is perpendicular to both of the original vectors. The alternating unit tensor allows us to calculate the magnitude and direction of this new vector. Specifically, the alternating unit tensor is used to calculate the cross product of two vectors by multiplying the two vectors together, using the alternating unit tensor as the multiplier. The result is the cross product of the two vectors. Answer 4: We can use the equations div E=0, div H=0, curl E=- \partialH/\partial t , curl H= \partialE /\partialt to show that E and H satisfy \nabla2u= \partial2u/\partialt2. Taking the curl of the equation curl E=- \partialH/\partial t , we get \nabla2E= -\partial2H/ \partialt2. Similarly, taking the curl of the equation curl H= \partialE /\partialt , we get \nabla2H = \partial2E/ \partialt2. Adding these two equations together, we get \nabla2(