Gradient

$\nabla f:\text{scalar}\to \text{vector}={\partial }_i{f}_i{e}_i$

• Can be thought of as the vector pointing to the direction which would lead to fastest ascent/increase of $f$
• If $f$ takes in any input and outputs a scalar output, then $\nabla f$‘s codomain is in ${\mathbb{R}}^n$
• $|\nabla f|$ is the magnitude of the slope along the direction of fastest ascent
• $\nabla$ is the del operator

$\nabla \vec{F}:\text{vector}\to \text{tensor}={\partial }_i{F}_j$

• 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

$({\nabla }_{A}f(A){)}_{ij}={\partial }_{A_{ij}}f(A)$

• Gradient of a function $f:{\mathbb{R}}^{n\times m}\to \mathbb{R}$ with respect to the input $A$

$f(x,y{)}_{\perp }=\nabla (f(x,y)-z)$

• 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

$\nabla (kf)=k\nabla f$

• $k$ is a scalar
• $f$ is a function

$\nabla (fg)=f\nabla g+g\nabla f$