What are the operations on vector field A and how do I simplify the results?

In summary, Graham explains the notation and conventions used for differentiation and curl of a vector field, and gives an example problem.
  • #1
gcooke
5
0
Hello. I am stuck trying to find an understandable answer to this online:

Carry out the following operations on the vector field A reducing the results to their simplest forms:

a. (d/dx i + d/dy j + d/dz k) . (Ax i + Ay j + Ax k)
b. (d/dx i + d/dy j + d/dz k) x (Ax i + Ay j + Ax k)

I know this is a dot product and cross product thing, and I think that A should become "integral Anda" (div) but I'm not sure what to do with the curl for b.

I can't find this worked out online. Can anyone direct me to a source?

Thank you!
Graham.
 
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  • #2
The best to do when you get problems as this one, is to use the abstract index notation. your first expression is the divergence of a vectorfield, and the second one is the curl, of the same field. So I will show the problem, for an arbitrary [tex]\vec v [/tex] vector field, multiplied by some A scalar field. I will denote the differential operator as: [tex]\frac{\partial}{\partial x_j} \equiv \partial_j[/tex]. And we use the einstein summation convention. That is we sum automaticaly on double indices. So:

[tex]\text{div}(A\vec{v}=\partial_k(Av_k)=v_k\partial_k A + A\partial_k v_k =(\vec{v}\nabla)A + A \text{div}\vec{v}[/tex]
As we see the first part is the "substantive" part of the total time derivative, we use in hydrodynamics. The operator explicitly in cartesian coordinates:
[tex]\vec{v}\nabla = v_x\frac{\partial}{\partial x}+v_y\frac{\partial}{\partial y}+v_z\frac{\partial}{\partial z}[/tex]
Now in your problem: [tex]\vec{v}=(x,y,x)[/tex]. The A scalar field is not given explicitly. So we have for the divergence according to the previous:
[tex]\text{div}(A\vec{v})=x\frac{\partial A}{\partial x}+y\frac{\partial A}{\partial y}+x\frac{\partial A}{\partial z}+A\left(\frac{\partial x}{\partial x}+\frac{\partial y}{\partial y}+\frac{\partial x}{\partial z}\right)=x\frac{\partial A}{\partial x}+y\frac{\partial A}{\partial y}+x\frac{\partial A}{\partial z}+2A[/tex]

For the second part we have to calculate the curl. Using the same convention as above:

[tex](\text{curl}(A\vec{v}))_k=\epsilon_{klm}\partial_lAv_m=A\epsilon_{klm}\partial_l v_m + \epsilon_{klm}(\partial_lA)v_m=[/tex]

[tex]=(A\text{curl}\vec{v}+(\text{grad}A)\times\vec{v})_k
[/tex]

Where [tex]\epsilon_{klm}[/tex] is the three indice totaly antisymmetric tensor, the levi civita tensor.

So now using the given field:

[tex]\text{curl}(A\vec{v})=A(0\;,\;-1\;,\;0)+(\text{grad}A)\times\vec{v}[/tex]

And we are done.
 
  • #3
Thank you very much Thaakisfox!

G.
 

1. What are the basic operations on vector fields?

The basic operations on vector fields include addition, subtraction, scalar multiplication, and dot product.

2. How do I add two vector fields together?

To add two vector fields, simply add the corresponding components of each vector. For example, if A = (Ax, Ay) and B = (Bx, By), then A + B = (Ax + Bx, Ay + By).

3. How do I subtract one vector field from another?

To subtract one vector field from another, simply subtract the corresponding components of each vector. For example, if A = (Ax, Ay) and B = (Bx, By), then A - B = (Ax - Bx, Ay - By).

4. How do I simplify the dot product of two vector fields?

The dot product of two vector fields is simplified by taking the product of their corresponding components and then adding them together. For example, if A = (Ax, Ay) and B = (Bx, By), then A · B = Ax*Bx + Ay*By.

5. What is the purpose of simplifying the operations on vector fields?

Simplifying the operations on vector fields allows for a better understanding of the behavior and characteristics of the vector field. It also allows for easier calculations and analysis of the vector field in various applications such as physics, engineering, and mathematics.

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