Vector calculus

• A vector-valued function (or vector function) is a function (of one or more variables) whose range is a set of vectors.
  • Given $\mathbf{r}(t)=⟨x(t),y(t),z(t)⟩$, we define:
    • $\underset{t\to a}{\lim }\mathbf{r}(t)=⟨\underset{t\to a}{\lim }x(t),\underset{t\to a}{\lim }y(t),\underset{t\to a}{\lim }z(t)⟩$.
    • $\frac{d\mathbf{r}}{dt}={\mathbf{r}}^{\prime }(t)=\underset{h\to 0}{\lim }\frac{\mathbf{r}(t+h)-\mathbf{r}(t)}{h}$ (assume $\mathbf{r}$ is differentiable)
      • ${\mathbf{r}}^{\prime }(t)=⟨x^{\prime }(t),y^{\prime }(t),z^{\prime }(t)⟩$

Gradient

The gradient of f(x,y) = −(cos²x + cos²y)² as a projected vector field on the bottom plane. (source: commons.wikimedia.org)

• The gradient vector (or gradient) of a differentiable function $f:{\mathbb{R}}^2\to \mathbb{R}$ is the vector-valued function $\nabla f:{\mathbb{R}}^2\to {\mathbb{R}}^2$ defined as $\nabla f=\left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y}\right)$, where $x,y$ are the variables of the function $f$.
  • $\nabla f(p_0)=\left(\frac{\partial f}{\partial x}(p_0),\frac{\partial f}{\partial y}(p_0)\right)$ is the gradient vector at the point $p_0$
  • $\frac{\partial f}{\partial \mathbf{v}}(p_0)=\nabla f(p_0)\cdot \mathbf{v}$
  • $\frac{\partial f}{\partial \hat{\mathbf{u}}}(p_0)=\nabla f(p_0)\cdot \hat{\mathbf{u}}=\Vert \nabla f(p_0)\Vert \cdot \cos \theta$, where $\hat{\mathbf{u}}$ is a unit vector and $\theta$ is the angle between $\nabla f(p_0)$ and $\hat{\mathbf{u}}$.
    • $\frac{\partial f}{\partial \hat{\mathbf{u}}}(p_0)$ is maximized when $\hat{\mathbf{u}}$ is in the direction of $\nabla f(p_0)$, minimized when $\hat{\mathbf{u}}$ is in the opposite direction, and zero when $\hat{\mathbf{u}}\perp \nabla f(p_0)$.

Divergence

• The divergence of a continuously differentiable vector field $\mathbf{F}=F_x\mathbf{i}+F_y\mathbf{j}+F_z\mathbf{k}$ is defined as the scalar-valued function $\mathrm{div}\mathbf{F}=\nabla \cdot \mathbf{F}=\left(\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}\right)\cdot (F_x,F_y,F_z)=\frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}$
• A vector field $\mathbf{F}$ is said to be solenoidal (or divergence-free or incompressible) if $\nabla \cdot \mathbf{F}=0$ everywhere

Curl

• The curl of a vector field $\mathbf{F}=F_x\mathbf{i}+F_y\mathbf{j}$ is defined as the vector-valued function $\text{curl}(\mathbf{F})=\nabla \times \mathbf{F}=\left(\frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y}\right)\mathbf{k}$.
• The curl of a vector field $\mathbf{F}=F_x\mathbf{i}+F_y\mathbf{j}+F_z\mathbf{k}$ is defined as the vector-valued function $\text{curl}(\mathbf{F})=\nabla \times \mathbf{F}=\left(\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z}\right)\mathbf{i}+\left(\frac{\partial F_x}{\partial z}-\frac{\partial F_z}{\partial x}\right)\mathbf{j}+\left(\frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y}\right)\mathbf{k}$.
• A vector field $\mathbf{F}$ is said to be irrotational (or curl-free) if $\nabla \times \mathbf{F}=0$ everywhere
• A vector field $\mathbf{F}:D\subseteq {\mathbb{R}}^n\to {\mathbb{R}}^n$ is said to be conservative if there exists a function $\varphi :D\to \mathbb{R}$ such that $\mathbf{F}=\nabla \varphi$. (such a function $\varphi$ is called a potential function for $\mathbf{F}$)
• Every conservative vector field is irrotational. The converse is true if the domain $D$ is simply connected.