- partial derivative
- vector field
- scalar field
- gradient
- divergence
- curl
- Laplacian
- line integral
- surface integral
- flux integral
- gradient theorem
- Gauss' theorem
- Stoke's theorem
- Green's theorem
- Hessian matrix
\[ \mathbf{H}_{f} \]
\[ (\mathbf{H}_{f})_{ij} = \frac{\partial^2 f}{\partial x_i \partial x_j} \]
- Jacobian matrix
\begin{definition}
The \emph{Jacobian matrix of $f$} is the matrix of first-order partial derivatives.
\[ \mathbf{J}_{\mathbf{F}} \]
\[ (\mathbf{J}_{\mathbf{F}})_{ij} = \frac{\partial \mathbf{F}_i}{\partial x_i} \]

- Multidimentional Taylor series
- Multidimentional Taylor's theorem
- Multidimensional first derivative test
- Multidimensional second derivative test

\begin{proposition}
Let $f$ have continuous second-order partial derivatives on $U$. Then $f$ is convex on $U$ iff its Hessian matrix $\mathbf{H}_{f}$ is positive semi-definite.
\end{proposition}

- Fubini's theorem
- Tonelli's theorem
- Clairaut's theorem
- directional derivative
- partial derivative
- vector-valued function
- mathematical-optimization: Lagrange multiplier theorem