curl v = omega cross r

[PG1995]

Hi

Problem #1: Could you please help me with this query?

Problem #2: Could you please help me with these queries too?

Regards
PG

 

[steveB]

Problem 1: I think you have it correct, but they are just trying to distinguish between angular and linear velocity, which differs in units as rad/s and m/s, for example.

Problem 2:

Q1: You are correct that there are two different operators. The Laplacian of a scalar is straightforward and used extensively. The vector-Laplacian is a stranger object and you need to be careful with it. The result they are getting is only obtained in rectangular coordinates, and in other coordinate systems, the result is more complicated and the wave equation form is not the standard one we are used to in basic physics. Both operators are second derivative types relative to position only. The time derivatives are written separately in vector math, and written explicitely.

Q2. As a result of the fact that the vector operations with the nabla sign (inverted Delta) are position derivatives, the time derivatives can be moved inside or outside as needed. This is only true in the differential (point form) equations. When integral forms of the equations are used, you have to consider that integral limits may be be time dependent also, and then you can't just move the time derivatives freely.

So, lets be clear. The fact that the partial with respect to time is an operator has nothing to do with it. In general, operators can not be rearranged unless they have special properties. In this case, the independence of time and position is the property that allows the differential operators to be moved.

 

[PG1995]

Laplacian operator

Hi

Could you please help me with these queries, Q1 and Q2? Thank you.

Regards
PG

 

[steveB]

For Q1, you have to be careful with operators and with notation in general. Don't ever let notation and symbology replace a mathematical proof. In general, the order of operators matter. For example, in matrix multiplications AB does not equal BA, in general. So, I don't think there is equality in this case, however, you are free to write it out and prove it for yourself. Remember that (A.grad)B is telling you to evaluate inside the parentheses first. This is important so that you end up with a vector, not a scalar. Also, since the order of operators matters, be careful and make sure the components of the A vector are in front of the partial derivatives from the gradient.

For Q2, remember that I mentioned before that the laplacian operator applied to a vector is a completely different animal than that applied to a scalar. So, your second method is correct. In rectangular coordinates ONLY, the laplacian of a vector equals the vector formed by the laplacian of each of the components, such that Laplacian(A)=i Laplacian(Ax) + j Laplacian(Ay) +k Laplacian (Az), which matches your second method. However, do not apply this in cylindrical or spherical coordinates. The form of the Laplacian of a vector should be looked up in those cases.