3 votes

Why is the derivative of the real absolute squared different to complex absolute squared

$$ \begin{align} \frac{\partial }{\partial x} |z|^2 &= \frac{\partial}{\partial x} (x^2 + y^2)\\ &= 2x \end{align} $$ No contradiction. The derivative of $|z|^2$ with respect to $x$ along the ...
2 votes

Prove that the following is incompressible (ie divergence is zero)

I am guessing that the quantity $Q= \vec r\cdot S \cdot\vec r$ is a scalar-valued expression, a quadratic function constructed from a symmetric matrix $S$ that has constant coefficients and no trace. ...
  • 2,073
2 votes
Accepted

Proof that linear combination of self adjoint maps is also self adjoint.

An operator $T:V\rightarrow V$ is self adjoint if $$\left<Tu,v\right> = \left<u,Tv\right>$$ for all $u,v\in V$. Fix $u,v\in V$ we want to prove that if $T$ and $S$ are self adjoint, then $$...
  • 12.5k
2 votes

Computing an integral related to Legendre polynomials

The following derivation is self-contained, in the sense that it does not leverage properties of Legendre polynomials. To facilitate notations, let $Q_n(X) = (X^2-1)^n$ and let $Q_n^{(n)}$ denote the $...
2 votes

Why isn't this divergence theorem question all equal to zero?

Your product rule is invalid. $u\cdot\sigma$ is contracting with one "slot" of $\sigma$, so $\nabla\cdot[u\cdot\sigma]$ is contracting with the other slot; you can't magically get a ...
2 votes
Accepted

Inner product is invariant under reflections in the real hyperplane?

Every vector $x$ can be decomposed into the sum of two components, $x_v=d(x,v)v$ and $x_{v^\perp}=x-d(x,v)v$, where $x_v$ is parallel to $v$, and $x_{v^\perp}$ is $d$-orthogonal to $v$ because $$ d(v,...
  • 127k
1 vote

Inner product is invariant under reflections in the real hyperplane?

$r_v(u) = 2d(u,v)v-u$, $$d(r_v(u),r_v(w)) = d(2d(u,v)v-u,2d(w,v)v-w)$$ $$ = -1 \times d(u,2d(w,v)v-w) + 2d(u,v) \times d(v,2d(w,v)v-w)$$ $$ = -1 \times (2d(u,v)\times d(w,v) - d(u,w)) + 2d(u,v) \...
  • 1,254
1 vote
Accepted

What is the operation that happens between a tensor product and a vector under matrix multiplication?

I would recommend Schuller's lecture on Multilinear Algebra for a very nice from-the-ground-up description of vector spaces and tensors, but it doesn't include a discussion of the tensor product. Lee'...
  • 258

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