6.4 KiB
Vector operators
Properties of the gradient, divergence and curl
Proposition: let
a,b \in \mathbb{R}
,f,g: \mathbb{R}^3 \to \mathbb{R}
be differentiable scalar fields and\mathbf{u}, \mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3
be differentiable vector fields. Then we have the following identities:Linearity:
\begin{align*} \nabla (af + bg) &= a \nabla f + b \nabla g, \ \nabla \cdot (a\mathbf{u} + b \mathbf{v}) &= a (\nabla \cdot \mathbf{u}) + b (\nabla \cdot \mathbf{v}), \ \nabla \times (a\mathbf{u} + b \mathbf{v}) &= a (\nabla \times \mathbf{u}) + b (\nabla \times\mathbf{v}). \end{align*}
Multiplication rules:
\begin{align*} \nabla (fg) &= f \nabla g+ g \nabla f, \ \nabla \cdot (f \mathbf{u}) &= f (\nabla \cdot \mathbf{u}) + \langle \nabla f, \mathbf{u} \rangle, \ \nabla \cdot (\mathbf{u} \times \mathbf{v}) &= \langle \nabla \times \mathbf{u}, \mathbf{v} \rangle - \langle \mathbf{u}, \nabla \times \mathbf{v} \rangle, \ \nabla \times (f\mathbf{u}) &= f (\nabla \times \mathbf{u}) + \nabla f \times \mathbf{u}. \end{align*}
??? note "Proof:"
Will be added later.
The laplacian
Definition: the laplacian of a differentiable scalar field
f: \mathbb{R}^3 \to \mathbb{R}
is defined as
\nabla^2 f(\mathbf{x}) := \nabla \cdot \nabla f(\mathbf{x}),
for all
\mathbf{x} \in \mathbb{R}^3
.
The notation may be unorthodox for some. An alternative notatation for the laplacian is \Delta f
, though generally deprecated.
We can also rewrite the laplacian for curvilinear coordinate systems as has been done below.
Theorem: the laplacian of a differentiable scalar field
f: \mathbb{R}^3 \to \mathbb{R}
for a curvilinear coordinate system is given by
\nabla^2 f(\mathbf{x}) = \frac{1}{g(\mathbf{x})} \partial_i \Big(\sqrt{g(\mathbf{x})} g^{ij}(\mathbf{x}) \partial_j f(\mathbf{x}) \Big),
for all
\mathbf{x} \in \mathbb{R}^3
.
??? note "Proof:"
Will be added later.
The laplacian for a ortho-curvilinear coordinate system may also be derived and can be found below.
Corollary: the laplacian of a differentiable scalar field
f: \mathbb{R}^3 \to \mathbb{R}
for a ortho-curvilinear coordinate system is given by
\nabla^2 f(\mathbf{x}) = \frac{1}{h_1 h_2 h_3} \bigg(\partial_1 \Big(\frac{h_2 h_3}{h_1} \partial_1 f(\mathbf{x}) \Big) + \partial_2 \Big(\frac{h_1 h_3}{h_2} \partial_2 f(\mathbf{x}) \Big) + \partial_3 \Big(\frac{h_1 h_2}{h_3} \partial_3 f(\mathbf{x}) \Big) \bigg),
for all
\mathbf{x} \in \mathbb{R}^3
.
??? note "Proof:"
Will be added later.
Please note that the scaling factors may also depend on the position \mathbf{x} \in \mathbb{R}^3
depending on the coordinate system.
Proposition: the laplacian of a differentiable vector field
\mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3
is given by
\nabla^2 \mathbf{v}(\mathbf{x}) = \nabla \big(\nabla \cdot \mathbf{v}(\mathbf{x})\big) - \nabla \times \big(\nabla \times \mathbf{v}(\mathbf{x})\big),
for all
\mathbf{x} \in \mathbb{R}^3
.
??? note "Proof:"
Will be added much later.
Potentials
Definition: a vector field
\mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3
is irrotational or curl free if
\nabla \times \mathbf{v}(\mathbf{x}) = \mathbf{0},
for all
\mathbf{x} \in \mathbb{R}^3
.
If \mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3
is the gradient of some scalar field \Phi: \mathbb{R}^3 \to \mathbb{R}
it is irrotational since
\nabla \times\big (\nabla \Phi(\mathbf{x})\big) = \mathbf{0},
for all \mathbf{x} \in \mathbb{R}^3
.
Proposition: an irrotational vector field
\mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3
has a scalar potential\Phi: \mathbb{R}^3 \to \mathbb{R}
such that
\mathbf{v}(\mathbf{x}) = \nabla \Phi(\mathbf{x}),
for all
\mathbf{x} \in \mathbb{R}^3
.
??? note "Proof:"
Will be added later.
In physics the scalar potential is generally given by the negative of the gradient, both are correct but one is more stupid than the other.
Definition: a vector field
\mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3
is solenoidal or divergence-free if
\nabla \cdot \mathbf{v}(\mathbf{x}) = 0,
for all
\mathbf{x} \in \mathbb{R}^3
.
If \mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3
is the curl of some vector field \mathbf{u}: \mathbb{R}^3 \to \mathbb{R}^3
it is solenoidal since
\nabla \cdot \big(\nabla \times \mathbf{u}(\mathbf{x}) \big) = 0,
for all \mathbf{x} \in \mathbb{R}^3
.
Proposition: a solenoidal vector field
\mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3
has a vector potential\mathbf{u}: \mathbb{R}^3 \to \mathbb{R}^3
such that
\mathbf{v}(\mathbf{x}) = \nabla \times \mathbf{u}(\mathbf{x}),
for all
\mathbf{x} \in \mathbb{R}^3
.
??? note "Proof:"
Will be added later.
The theorem below is the Helmholtz decomposition theorem and states that every vector field can be written in terms of two potentials.
Theorem: every vector field
\mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3
can be written in terms of a scalar\Phi: \mathbb{R}^3 \to \mathbb{R}
and a vector\mathbf{u}: \mathbb{R}^3 \to \mathbb{R}^3
potential as
\mathbf{v}(\mathbf{x}) = \nabla \Phi(\mathbf{x}) + \nabla \times \mathbf{u}(\mathbf{x}),
for all
\mathbf{x} \in \mathbb{R}^3
.
??? note "Proof:"
Will be added later.
It then follows that the scalar and vector potentials can be determined for a volume V \subset \mathbb{R}^3
with a boundary surface A \subset \mathbb{R}^3
that encloses the domain V
.
Corollary: the scalar
\Phi: \mathbb{R}^3 \to \mathbb{R}
and vector\mathbf{u}: \mathbb{R}^3 \to \mathbb{R}^3
potentials for a volumeV \subset \mathbb{R}^3
with a boundary surfaceA \subset \mathbb{R}^3
that encloses the domainV
are given by
\begin{align*} \Phi(\mathbf{x}) &= \frac{1}{4\pi} \int_V \frac{\nabla \cdot \mathbf{v}(\mathbf{r})}{|\mathbf{x} - \mathbf{r}|}dV - \frac{1}{4\pi} \oint_A \bigg\langle \frac{1}{|\mathbf{x} - \mathbf{r}|} \mathbf{v}(\mathbf{r}), d\mathbf{A} \bigg\rangle, \ \ \mathbf{u}(\mathbf{x}) &= \frac{1}{4\pi} \int_V \frac{\nabla \times \mathbf{v}(\mathbf{r})}{|\mathbf{x} - \mathbf{r}|}dV - \frac{1}{4\pi} \oint_A \frac{1}{|\mathbf{x} - \mathbf{r}|} \mathbf{v}(\mathbf{r}) \times d\mathbf{A}, \end{align*}
for all
\mathbf{x} \in \mathbb{R}^3
.
??? note "Proof:"
Will be added later.