Gradient
#Math
$\displaystyle \nabla f:\text{scalar}\rightarrow \text{vector}=\partial_{i}f_{i}e_{i}$
- Can be thought of as the vector pointing to the direction which would lead to fastest ascent/increase of $\displaystyle f$
- If $\displaystyle f$ takes in any input and outputs a scalar output, then $\displaystyle \nabla f$'s codomain is in $\displaystyle \mathbb{R}^{n}$
- $\displaystyle \lvert \nabla f\rvert$ is the magnitude of the slope along the direction of fastest ascent
- $\displaystyle \nabla$ is the del operator
$\displaystyle \nabla \vec{F}:\text{vector}\rightarrow \text{tensor}=\partial_{i}F_{j}$
- Decompose into
- Trace of matrix is divergence/how volume changes
- Anti-symmetric tensor is curl/how rotation changes without volume or shape change
- Traceless symmetric tensor is shear/how deformation occurs without volume change
$\displaystyle (\nabla {A}f(A)){ij}=\partial_{A_{ij}}f(A)$
- Gradient of a function $\displaystyle f:\mathbb{R}^{n\times m}\rightarrow \mathbb{R}$ with respect to the input $\displaystyle A$
$\displaystyle f(x,y)_{\perp}=\nabla (f(x,y)-z)$
- The normal vector to a multivariable function which may represent a surface can be found by taking the gradient of (the function - an extra dimensional variable)
- Desmos Demo
Properties
$\displaystyle \nabla(kf)=k\nabla f$
- $\displaystyle k$ is a scalar
- $\displaystyle f$ is a function