---
date: '2024-11-27'
description: just enough vector calculus to be dangerous
id: Vector calculus
modified: 2026-06-05 15:08:26 GMT-04:00
tags:
- math
title: Vector calculus
created: '2024-11-27'
published: '2024-11-27'
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slug: thoughts/Vector-calculus
permalink: https://aarnphm.xyz/thoughts/Vector-calculus.md
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---
## divergence
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.
> \[!definition\] Definition 1.
>
> 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
$$
\operatorname{div} \mathbf{F} \big|_{\mathbf{x}_0} = \lim_{V \to 0} \frac{1}{|V|} \oiint_{S(V)} \mathbf{F} \cdot \hat{\mathbf{n}} \, dS
$$
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.
### Cartesian coordinates
for a continuously differentiable vector field $\mathbf{F} = F_x \mathbf{i} + F_y \mathbf{j} + F_z \mathbf{k}$, divergence is defined as the scalar-valued function:
$$
\begin{aligned}
\operatorname{div} \mathbf{F} = \nabla \cdot \mathbf{F} &= \left( \frac{\partial}{\partial{x}}, \frac{\partial}{\partial{y}}, \frac{\partial}{\partial{z}} \right) \cdot \left( F_x, F_y, F_z \right) \\
&=\frac{\partial{F_x}}{\partial{x}} + \frac{\partial{F_y}}{\partial{y}} + \frac{\partial{F_z}}{\partial{z}}
\end{aligned}
$$
## Jacobian matrix
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:
$$
\begin{equation}
\begin{aligned}
\mathbf{J}_{\mathbf{f}}
&= \begin{bmatrix}
\frac{\partial \mathbf{f}}{\partial x_1} & \cdots & \frac{\partial \mathbf{f}}{\partial x_n}
\end{bmatrix} \\
&= \begin{bmatrix}
\nabla^T f_1 \\
\vdots \\
\nabla^T f_m
\end{bmatrix} \\
&= \begin{bmatrix}
\frac{\partial f_1}{\partial x_1} & \cdots & \frac{\partial f_1}{\partial x_n} \\
\vdots & \ddots & \vdots \\
\frac{\partial f_m}{\partial x_1} & \cdots & \frac{\partial f_m}{\partial x_n}
\end{bmatrix}.
\end{aligned}
\end{equation}
$$
> \[!definition\] Definition 2. Jacobian determinant
>
> When $m = n$, the Jacobian matrix is a square, so its determinant is a well-defined function of $x$ [^conjecture]
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 [[thoughts/Vector calculus#gradient]]== of $f$, or $\mathbf{J}_f = \nabla^T f$
Intuitively, one can think about Jacobian matrix as the linear map of a function onto a subspace.
[^conjecture]: See also [Jacobian conjecture](https://en.wikipedia.org/wiki/Jacobian_conjecture)
## gradient
see also [[thoughts/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:
$$
df = \nabla f \cdot d \mathbf{r}
$$
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 $\nabla f$
> \[!definition\] Definition 3.
>
> 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: [^grad-annotation]
>
> $$
> (\nabla f(x)) \cdot \mathbf{v} = D_v f(x)
> $$
source code
[^grad-annotation]: another annotation often used in [[thoughts/Machine learning]] is `grad(f)`. See also [[thoughts/Automatic Differentiation|autograd]]
## vector field
> an assignment of vector to each point in a space, most commonly Euclidean space $\mathbb{R}^n$
> \[!definition\] Definition 4. 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 $S$
