Vector calculus

operates on vector field producing a scalar field giving quantity of the gector field’s source at each points.

represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.

definition

the divergence of a vector field $\mathbf{F}(\mathbf{x})$ at point $x_{0}$ is defined as the limit of the ratio of the surface integral of $\mathbf{F}$ out of the closed surface of a volume $V$ enclosing $x_0$ to the volume of $V$, as $V$ shrinks to zero

where $|V|$ is the volume of $V$, $S(V)$ is the boundary of $V$ and $\hat{\mathbf{n}}$ is the outward unit normal to that surface.

Suppose a function $\mathbf{f}: \mathbf{R}^n \to \mathbf{R}^m$ is a function such that each of its first-order partial derivatives exists on $\mathbf{R}^n$, then the Jacobian matrix of $\mathbf{f}$ is defined as follows:

Jacobian determinant

When $m = n$, the Jacobian matrix is a square, so its determinant is a well-defined function of $x$ ¹

When $m=1$, or $f: \mathbf{R}^n \to \mathbf{R}$ is a scalar-valued function, then Jacobian matrix reduced to the row vector $\nabla^T f$, and this row vector of all first-order partial derivatives of $f$ is the transpose of the gradient of $f$, or $\mathbf{J}_f = \nabla^T f$

see also gradient descent

a vector field $\nabla f$ whose value at a point $p$ gives the direction and the rate of fastest increase.

In coordinate-free term, the gradient of a function $f(\mathbf{r})$ maybe defined by:

where $df$ is the infinitesimal change in $f$ for an infinitesimal displacement $d \mathbf{r}$, and is seen to be maximal when $d \mathbf{r}$ is in the direction of the gradient $\nabla f$

definition

the gradient of $f$ ($\nabla f$) is defined as the unique vector field whose dot product with any vector $\mathbf{v}$ at each point $x$ is the directional derivative of $f$ along $\mathbf{v}$, such that: ²

$$(\nabla f(x)) \cdot \mathbf{v} = D_v f(x)$$

an assignment of vector to each point in a space, most commonly Euclidean space $\mathbb{R}^n$

vector fields on subsets of Euclidean space

Given a subset of $S$ of $\mathbb{R}^n$, a vector field $V: S \to \mathbb{R}^n$ in standard Cartesian coordinates $(x_{1},\ldots,x_{n})$.

For every smooth vector field $V$ in an open subset $S$ of $\mathbb{R}^n$ can be written as:

$$\sum_{i=1}^{n} V_i (x_{1},\ldots,x_{n}) \frac{\partial}{\partial x_i}$$

For some smooth functions $V_{1},\ldots,V_{n}$ on $S$