Evaluate the partial derivative of a matrix element

[Biffinator87]

Homework Statement

A determinant a is defined in the following manner a_r * A^k = Σⁿ_s=1 a_rs A^ks = δ^k_r a , where a=det(a_ij), a_r , A_k , are rows of the coefficient matrix and cofactor matrix respectively. The second term in the equation is the expansion over the columns of both matrices, δ^k_r is the kronecker delta that is 1, when r=k and 0 otherwise. Or a_r * A^r = Σⁿ_s=1 a_rs A^rs = a, where a=det(a_ij). Evaluate (∂a / ∂a_rt), where a_rt is the coefficient matrix element in row r and column t. Assume all of the elements of the coefficient matrix depend on the independent variables (u₁,u₂,u₃). Find (∂a / ∂u_t) for t=1,2,3. Your answer should depend on your results for (∂a / ∂a_rt) above.

Homework Equations

No equations given. We are going over the basics of curvilinear coordinates at the moment and thus I believe tangent vector equations are need to solve the problem which is what I think he wants us to find when he says evaluate (see above). However, the professor is very unhelpful and not willing to really help the students understand exactly what he is after. To be fair he has had some kindof illness that has left him partially disabled.

The Attempt at a Solution

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My attempt at the solution poses a problem. The professor has not given any direction as to how to approach these matrix problems. My only attempt that the solution so far was to use the determinant to find the matrix element a_rt and then from there right a very generic answer for what that vector would be in terms of the independent variables given: a_rt = u₁ i + u₂ j + u₃ k. the problem I am encountering is I just can't seem to figure out if this is the correct route to take. We are using a schaums outline for the book and I haven't found any examples in there to help. Any direction that can be offered would be greatly appreciated.

If i have violated the rules of the forums please reply and let me know that I have so that I can correct myself appropriately. I understand we are supposed to have a valid attempt that the problem before posting. I hope I explained clearly enough above that I have tried to figure out what I need to do.

Thanks!

 

[andrewkirk]

You have the equation $\sum_{s=1}^n a_{rs}A^{ks}=\delta_r^ka$. If you set $k=r$ then you get the equation $a=\sum_{s=1}^n a_{rs}A^{rs}=$. Note that none of the items $A^{rs}$ depend on any coefficient in row $r$, and only one of the $a_{rs}$ coefficients in that sum will be the item with respect to which you are asked to differentiate: $a_{rt}$. What do you get when you take the $\frac{\partial}{\partial a_{rt}}$ of the sum?

When you get to the second part, I suggest you take $\frac{\partial a}{\partial u_j}$ rather than $\frac{\partial a}{\partial u_t}$ as $t$ is already used for something completely different, and using the same index for two different things is unnecessarily confusing.

 

[Biffinator87]

You have the equation $\sum_{s=1}^n a_{rs}A^{ks}=\delta_r^ka$. If you set $k=r$ then you get the equation $a=\sum_{s=1}^n a_{rs}A^{rs}=$. Note that none of the items $A^{rs}$ depend on any coefficient in row $r$, and only one of the $a_{rs}$ coefficients in that sum will be the item with respect to which you are asked to differentiate: $a_{rt}$. What do you get when you take the $\frac{\partial}{\partial a_{rt}}$ of the sum?

When you get to the second part, I suggest you take $\frac{\partial a}{\partial u_j}$ rather than $\frac{\partial a}{\partial u_t}$ as $t$ is already used for something completely different, and using the same index for two different things is unnecessarily confusing.

Thank you! I will try it and see what I come up with!