Help Interpeting Mathematica Result

[pholvey]

Hi everyone,

Given the attached input and result, I'm confused as to how to interpret the 1.1, x.1, and y.1 terms in the solution. Does this mean I'm supposed to dot 1 into 1? How does that work? Any help is appreciated. Of course, x and y are vectors (not that it matters but they're xyz coordinates). Thanks for any help!

pholvey

 

[Simon_Tyler]

Do you mean to have a vector derivative or are x and y scalars (normal variables)?

If you don't, then remove the Dot[] - ie type "x y" or "x*y" instead of "x.y".

If you do mean to have symbolic vector derivatives, then you're in trouble, because Mathematica does't know how to do that for arbitrary n-dimensional vectors.

 

[pholvey]

Hey Simon,

Ya, I did mean to have symbolic vector derivatives. Unfortunate that Mathematica doesn't know how to handle them...

 

[pholvey]

Just for kicks though, how would you go about working out the derivative of the equation as it is shown in the attachment?

 

[Simon_Tyler]

Well, if you interpret the 1's as Identity matrices / Kronecker Delta's, then it's basically correct.

Let's drop the extranious stuff and just take the derivative
ans = ∂_x∂_y(x.y + c)²
where ∂_x is the partial derivative with respect to x and c is a constant.

So,
ans_ij = ∂_xi∂_yj(x.y + c)²
ans_ij = 2 ∂_xi(x_j(x.y + c))
ans_ij = 2 δ_ij(x.y + c) + 2 y_i x_j

Written in vector/matrix form this is
ans = 2 I (x.y+c) + 2 y x^T
where I is the identity matrix and y x^T is one way of writing the outer product.

You can check this in Mathematica for any particular dimension using something like

In[1]:= With[{n = 14},
          X = Array[x, {n}];
          Y = Array[y, {n}];
          II = IdentityMatrix[n];]

In[2]:= D[(X.Y + c)^2, {X}, {Y}] == 2 (II (X.Y + c) + {Y}\[Transpose].{X}) // Expand

Out[2]= True