Vector

• Vector 16 Dec 2020 21:06 UTC

  3 points

  on: A Primer on Ma­trix Calcu­lus, Part 2: Ja­co­bi­ans and other fun

  3blue1brown touches on that way of thinking about functions/​derivatives in this video

  (very similar to Khan academy video linked in the article)
  

• Vector 16 Dec 2020 20:05 UTC

  1 point

  on: A Primer on Ma­trix Calcu­lus, Part 1: Ba­sic review

  In this paragraph:

  >In single variable calculus, the chain rule can be written as $\frac{d}{dx}f\left(g\right(x\left)\right)=f^{\prime }\left(g\right(x\left)\right)g^{\prime }\left(x\right)$. In the multivariable case, for a function $f:R^k\to R$, we write $\frac{d}{dx}f\left(g\right(x\left)\right)={\sum }_{i=1}^k\frac{d}{dx}{g}_i\left(x\right)D_{i}f\left(g\right(x\left)\right)$ where $D_i$ is the partial derivative of $f$ with respect to its $i$th argument. This can be simplified by employing the following notation, which uses a dot product: $\frac{d}{dx}f\left(g\right(x\left)\right)=\nabla f\cdot g^{\prime }\left(x\right)$.

  What does $g_i\left(\right)$ stand for?
  My understanding that g maps $g:R^m\to R^k$ and $g_i$ indicates i-th element of a vector that g produces.
  

  Is that the right way to think about it?