Vector Analysis

This page is a sub-page of our page on Calculus of Several Real Variables.

///////

The interactive simulations on this page can be navigated with the Free Viewer
of the Graphing Calculator.

///////

Related KMR-pages:

Gradients
Linear Algebra

Exact Differential Forms
Partial Differential Equations

Complex Derivative
Complex trigonometry
Conformal Mapping
Inversion
Möbius transformations
Conformal Mapping
Stereographic Projection
The Riemann Zeta function
Einstein for Flatlanders

In Swedish:

Vektoranalys

///////

Books:

In Swedish:

• Ramgard, A., Vektoranalys – 2:a upplagan,
Teknisk Högskolelitteratur i Stockholm AB (THS AB), 1992.

/////// Translating from Ramgard (1992, page 1):

1.1 Vector-valued functions

A vector-valued function \, \textbf{A} \, is a function whose codomain \, B \, consists of vectors. Let us assume that \, \textbf{A} :s domain \, D \, consists of \, n -tuples \, (u, v, \cdots) \, of real numbers. In that case the function \, \textbf{A} \, uniquely associates a vector \, \textbf{A} (u, v, \cdots) \, with every set of values of the independent variables \, u, v, \cdots \, that corresponds to a point in \, D \,

In general, we will consider vectors that belong to a three-dimensional vector space and therefore can be represented by arrows in the “usual” three-dimensional space \, \mathbb{E}^3 . We often use cartesian coordinates \, x, y, z \, in order to label the points in \, \mathbb{E}^3 . By a cartesian coordinate system we always mean an orthogonal and right-handed system.

An arbitrary vector \, \textbf{A} \, can be referred to the basis-vectors \, {\textbf{e}}_x, {\textbf{e}}_y, {\textbf{e}}_z \, in the cartesian coordinate system:

\textbf{A} \, = \, (A_x, A_y, A_z) \, \equiv \, A_x {\textbf{e}}_x + A_y {\textbf{e}}_y + A_z {\textbf{e}}_z. \qquad \qquad \qquad \qquad \qquad (1.1)

According to (1.1) a vector-valued function is equivalent to three real-valued functions:

\, A_x(u, v, \cdots), A_y(u, v, \cdots), A_z(u, v, \cdots) .

Definition: A vector-valued function \textbf{A} (u, v, \cdots) \, is said to be continuous at the point \, (u, v, \cdots) \, if, for each value of \, \epsilon > 0 , one can find a \, \delta(\epsilon) > 0 \, such that

\, 0 < | \, \Delta u \, | < \delta \, , \, 0 < | \, \Delta v \, | < \delta \, , \, \cdots \, \implies

| \, \textbf{A}(u + \Delta u, v + \Delta v, \cdots) \, - \, \textbf{A} (u, v, \cdots) \, | \, < \, \epsilon .

\textbf{A}(u, v, \cdots) \, is continuous if and only if
the component functions \, A_x(u, v, \cdots), \cdots \, are continuous functions.

Definition: A vector-valued function \textbf{A} (t) \, has the limit \textbf{A} (t_0) \, when \, t \, tends to \, t_0 \, :

\lim\limits_{t \rightarrow t_0} \, \textbf{A} (t) \, = \, \textbf{A} (t_0) , if, for each value of \, \epsilon > 0 , there exists a \, \delta(\epsilon) > 0 \, such that

0 < | \, t - t_0 \, | < \delta \, \implies \, | \, \textbf{A} (t) - \, \textbf{A} (t_0) \, | \, < \epsilon .

/////// End of the translation from from Ramgard (1992).

/////// Translating from Ramgard (1992, page 7):

2.1 Differentiation and integration of vector-valued functions

Derivatives of vector-valued functions are formally defined in the same way as derivatives of scalar-valued functions:

Definition: Let \textbf{A} (u) \, be a vector-valued function, and let

\, \Delta \textbf{A} \, \equiv \, \textbf{A}(u + \Delta u) - \textbf{A}(u) .

If the limit \, \lim\limits_{t \rightarrow t_0} \, \frac{\Delta \textbf{A}}{\Delta u} \, , exists,
the function \textbf{A} (u) \, is said to have the derivative \, \frac{d \textbf{A}}{du} \stackrel {\mathrm{def}}{=} \lim\limits_{\Delta u \rightarrow 0} \, \frac{\Delta \textbf{A}}{\Delta u} .

\, \frac{d \textbf{A}}{du} \, , which is computed as the limit of the quotient between a vector and a scalar, is obviously also a vector-valued function. If \, \frac{d \textbf{A}}{du} \, is differentiated we get the second derivative \, \frac{d^2 \textbf{A}}{du^2} \, etc.

We get a geometric interpretation of the derivative if we lay out all the vectors \, \textbf{A} (u) \, from a common point \, O . Then the tips of these vectors trace out a curve in space,
the so-called hodograph, and \, \frac{d \textbf{A}}{du} \, is a tangent vector to this curve.

The cartesian components of the derivative of a vector
are computed by differentiation of the cartesian components of the vector.

Theorem 2.1 \,\, \frac{d \textbf{A}}{du} = \frac{d}{du} (A_x, A_y, A_z) = ( \frac{d \textbf{A}_x}{du}, \frac{d \textbf{A}_y}{du}, \frac{d \textbf{A}_z}{du} ) .

Proof: By component-wise differentiation (see page 8). \qquad \qquad \qquad \qquad \qquad \boxdot

One can also let theorem 2.1 be the definition of \, \frac{d \textbf{A}}{du} .

Theorem 2.2 Let \, \textbf{A} (u) \, och \, \textbf{B} (u) \, be vector-valued functions
and let \, \Phi (u) \, be a scalar-valued function.
Then the following differentiation rules apply:

\,\frac{d}{du}(\textbf{A} + \textbf{B}) = \frac{d \textbf{A}}{du} + \frac{d \textbf{B}}{du} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (2.3)

\,\frac{d}{du}(\textbf{A} \cdot \textbf{B}) = \frac{d \textbf{A}}{du} \cdot \textbf{B} + \textbf{A} \cdot \frac{d \textbf{B}}{du} \qquad \qquad \qquad \qquad \qquad \qquad \quad \qquad \;\, (2.4)

\,\frac{d}{du}(\textbf{A} \times \textbf{B}) = \frac{d \textbf{A}}{du} \times \textbf{B} + \textbf{A} \times \frac{d \textbf{B}}{du} \qquad \qquad \qquad \qquad \qquad \quad \qquad \;\;\; (2.5)

\,\frac{d}{du}(\Phi \textbf{A}) = \frac{d \Phi}{du}\textbf{A} + \Phi \frac{d \textbf{A}}{du}. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \;\; (2.6)

Proof: [The proofs of these differentiation rules] are formally identical to the proofs of the corresponding differentiation rules for real-valued functions. This is because the proofs only make use of arithmetical laws - the commutativity of addition and the distributivity of multiplication w.r.t. addition - which hold for vectors as well as for scalars. \qquad \quad \boxdot

Alternatively, one can prove (2.3 - 2.6) by making use of theorem 2.1 and the component representations of \, \textbf{A} + \textbf{B} \, , \, \textbf{A} \cdot \textbf{B} \, , \, \cdots

[...]

Theorem 2.3 Assume that \, \textbf{A} = \textbf{A}(u) \, and that \, u = u(v) \, are differentiable functions of \, u \, respectively \, v . Then we have

\, \frac{d}{dv} \textbf {A}(u(v)) \, = \, \frac{d \textbf{A}}{du} \frac{du}{dv}. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \;\;\; (2.7)

Proof: Make use of theorem 2.1 and the chain rule for real-valued functions. \qquad \boxdot

2.2 Partial derivatives of vector-valued functions

Definition: By the partial derivative of \, A(u, v, ...) \, with respect to \, u \,
we mean the following limit (assuming that it exists):

\, \frac{\partial \textbf{A}}{\partial u} \, = \, \lim\limits_{\Delta u \rightarrow 0} \, \frac{ \textbf{A}(u + \Delta u, v, \cdots) \, - \, \textbf{A}(u, v, \cdots) }{\Delta u} .

Theorem 2.4 \,\, \frac{\partial \textbf{A}}{\partial u} = \frac{ \partial }{\partial u} (A_x, A_y, A_z) = ( \frac{\partial \textbf{A}_x}{\partial u}, \frac{\partial \textbf{A}_y}{\partial u}, \frac{\partial \textbf{A}_z}{\partial u} ) .

Theorem 2.5 If \, \textbf{A}(u, v, \cdots) \, och \, \textbf{B} (u, v, \cdots) \, are differentiable, vector-valued
functions, and if \, \Phi (u, v, \cdots) \, is a differentiable, scalar-valued function, the following partial differentiation rules apply:

\,\frac{\partial}{\partial u}(\textbf{A} + \textbf{B}) = \frac{\partial \textbf{A}}{\partial u} + \frac{\partial \textbf{B}}{\partial u} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (2.10)

\,\frac{\partial}{\partial u}(\textbf{A} \cdot \textbf{B}) = \frac{\partial \textbf{A}}{\partial u} \cdot \textbf{B} + \textbf{A} \cdot \frac{\partial \textbf{B}}{\partial u} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \;\, (2.11)

\,\frac{\partial}{\partial u}(\textbf{A} \times \textbf{B}) = \frac{\partial \textbf{A}}{\partial u} \times \textbf{B} + \textbf{A} \times \frac{\partial \textbf{B}}{\partial u} \qquad \qquad \qquad \qquad \qquad \qquad \quad \;\;\;\, (2.12)

\,\frac{\partial}{\partial u}(\Phi \textbf{A}) = \frac{\partial \Phi}{\partial u}\textbf{A} + \Phi \frac{\partial \textbf{A}}{\partial u}. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \;\;\, (2.13)

/////// End of the translation from from Ramgard (1992).

/////// Translating from Ramgard (1992, page 11):

2.3 Differentials of vector-valued functions

Let \, \textbf{A}(u, v, \cdots) \, be a vector-valued function
whose partial derivatives \, \partial \textbf{A} / \partial u , \, \partial \textbf{A} / \partial v , \, \cdots \, are continuous functions.

We introduce the change of the function value

\, \Delta \textbf{A} \equiv \textbf{A}(u + \Delta u, v + \Delta v, \, \cdots) \, \qquad \qquad \qquad \qquad \qquad \qquad \qquad \;\; (2.17)

and the differential of \, \textbf{A} \,

\, d \textbf{A} \, \equiv \, \frac{\partial \textbf{A}}{\partial u} du \, + \, \frac{\partial \textbf{A}}{\partial v} dv \, + \, \cdots \, . \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \; (2.18)

Theorem 2.7 The change \, \Delta \textbf{A} \, is approximated arbitrarily close by the differential \, d \textbf{A} \,
in the sense that

\, \Delta \textbf{A} \, = \, d \textbf{A} + \textbf{h} du + \textbf{k} dv + \cdots \, . \qquad \qquad \qquad \qquad \qquad \qquad \qquad \; (2.19)

where \, \textbf{h} \, och \, \textbf{k} \, are vectors whose lengths go to zero when \, d u \, , \, d v \, , \, \cdots \, goes to zero.

[...]

Theorem 2.8 If \, \textbf{A} \, , \, \textbf{B} \, and \, \Phi \, are differentiable functions,
the following rules for differentiation apply:

\, d (\textbf{A} + \textbf{B}) \, = \, d \textbf{A} + d \textbf{B} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (2.20)

\, d (\textbf{A} \cdot \textbf{B}) \, = \, d \textbf{A} \cdot \textbf{B} + \textbf{A} \cdot d \textbf{B} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \;\, (2.21)

\, d (\textbf{A} \times \textbf{B}) \, = \, d \textbf{A} \times \textbf{B} + \textbf{A} \times d \textbf{B} \qquad \qquad \qquad \qquad \qquad \qquad \quad \;\;\;\, (2.22)

\, d (\Phi \textbf{A}) \, = \, d \Phi \textbf{A} + \Phi d \textbf{A}. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \;\;\, (2.23)

[...]

2.4 The differential of the position vector

The position vector \, \textbf{r} \, from the origin \, O \, to a point \, P \,
can be viewed as a function of \, P :s cartesian coordinates \, x, y, z \, :

\, \textbf{r} \, = \, \textbf{r}(x, y, z) \, = \, x {\textbf{e}}_x + y {\textbf{e}}_y + z {\textbf{e}}_z \, = \, (x, y, z). \qquad \qquad \qquad \qquad \; (2.24)

The cartesian components of the differential of the position vector \, \textbf{r}
can be obtained as the differentials of the cartesian components of \, \textbf{r} :

\, d \textbf{r} \, = \, (d x, d y, d z). \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \, (2.25)

\, d \textbf{r} \, approximates the change \, \Delta \textbf{r} \, of the position vector
when moving from the point \, P : x, y, z \,
to a neighboring point \, P' : x + d x, y + d y, z + d z .

In this special case there is exact equality,
since the function (2.24) is linear in the independent variables.

/////// End of the translation from from Ramgard (1992).

/////// Translating from Ramgard (1992, page 19):

3.1 The gradient and the directional derivative

Let \, \Phi \, be a continuously differentiable scalar field, by which we mean
(and will mean below) that the three partial (first) derivatives are continuous functions.

At the point \, \textbf{r}(x, y, z) \, the field assumes the value \, \Phi(x, y, z) \,
and at the neighboring point \, \textbf{r} + d \textbf{r} = (x + d x, y + d y, z + d z) \,
the field assumes the value \, \Phi(x, y, z) + \Delta \Phi , where

\, \Delta \Phi \, \approx \, d \Phi \, = \, \frac{\partial \Phi}{\partial x} d x + \frac{\partial \Phi}{\partial y} d y + \frac{\partial \Phi}{\partial z} d z. \qquad \qquad \qquad \qquad \qquad \qquad \;\; (3.1)

The partial derivatives in (3.1) are evaluated at the point \, \textbf{r} .

We now introduce a continuous vector field \, \text{grad} \, \Phi ,
which concisely describes \, \Phi :s variation in the immediate vicinity of each point:

Definition: The gradient of the scalar field \, \Phi \, is the vector field

\, \text{grad} \, \Phi \, \stackrel {\mathrm{def}}{=} \, (\frac{\partial \Phi}{\partial x}, \frac{\partial \Phi}{\partial y}, \frac{\partial \Phi}{\partial z}). \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (3.2)

Evidently, the differential of \, \Phi \, i (3.1) can be written as
the scalar product of \, \text{grad} \, \Phi \, and the differential of the position vector:

\, d \Phi = \text{grad} \, \Phi \cdot d \textbf{r}. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \;\; (3.3)

NOTE: The expression (3.3) can be used as a coordinate-free definition of the gradient
since it does not refer to any specific coordinate system in space.

We now introduce into equation (3.3) the modulus \, d s \,
and the direction unit-vector \, \textbf{e} \, of the position-vector differential \, d \textbf{r} \, :

\, d \textbf{r} = \textbf{e} \, d s \, \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \;\; (3.4)

Next we divide (3.4) by \, d s . In this way we arrive at
the directional derivative along the direction \, \textbf{e} \, away from the point \, \textbf{r} \, :

\, \frac{d \Phi}{d s} = \text{grad} \, \Phi \cdot \textbf{e}. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad (3.5)

The rate of increase of \, \Phi \, along a given direction \, \textbf{e} \, is therefore equal to
the component of the gradient vector \, \text{grad} \, \Phi \, along this direction.

If we want, we can define the directional derivative as:

\, \frac{d \Phi}{d s} = \lim\limits_{s \, \rightarrow \, 0} \frac{ \Phi ( \textbf{r} + s \textbf{e} ) - \Phi ( \textbf{r} ) }{ s }. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \;\; (3.6)

Theorem 3.1 The value of \, \text{grad} \, \Phi \, at the point \, P \, , the vector \, {(\text{grad} \, \Phi)}_P \, ,
points in the direction along which \, \Phi \, increases the fastest when moving away from \, P .
Moreover, the maximal increase of \, \Phi \, per unit of length is equal to \, | {(\text{grad} \, \Phi)}_P | .

Proof: The directional derivative along the direction \, \textbf{e} \, :

\, \frac{d \Phi}{d s} = \text{grad} \, \Phi \cdot \textbf{e} = | {(\text{grad} \Phi)}_P | \, \cos \alpha ,

has its maximum equal to \, | {(\text{grad} \, \Phi)}_P | \, when \, \alpha = 0 ,
that is, when \, \textbf{e} \, \shortparallel \, \text{grad} \, \Phi . \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \;\;\; \boxdot

Theorem 3.2 If \, \Phi \, has maximum or minimum at a point, then \, \text{grad} \, \Phi = 0 \, at this point.

Theorem 3.3 \, \text{grad} \, \Phi \, at the point \, P \, is orthogonal to the level surface \, \Phi = c \, that passes through the point \, P .

Proof:The value of the scalar field remains unchanged when the field is subjected to a small displacement \, d \textbf{r} \, along a level surface:

\, d \Phi = \text{grad} \, \Phi \cdot d \textbf{r} = 0 ,

which says that \, \text{grad} \, \Phi \, is orthogonal to each \, d \textbf{r} \, in the level surface,
which means that \, \text{grad} \, \Phi \, is orthogonal to the level surface. \, \qquad \qquad \qquad \quad \boxdot

Theorem 3.4 The perpendicular distance at the point \, P \,
between the closely situated level surfaces \, \Phi = c \, och \, \Phi = c + h \,
is approximately equal to:

\, \Delta s \approx \frac{ h }{ | {(\text{grad} \, \Phi)}_P | } \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \; (3.7)

Proof: Let \, d \textbf{r} \, i (3.3) be orthogonal to \, \Phi = c \, i.e., parallel to \, {(\text{grad} \, \Phi)}_P .
Moreover, let \, d\Phi \approx \Delta \Phi = h \, and \, | d \textbf{r} | \approx \Delta s . \qquad \qquad \qquad \qquad \qquad \quad \boxdot

The density of surfaces in the family of level surfaces \, \Phi = c + n h \, , \, n \in \mathbb{Z} \,
is therefore directly proportional to the modulus of the gradient vector.

/////// End of the translation from from Ramgard (1992).

/////// Translating from Ramgard (1992, page 22):

3.2 The potential

Definition: Consider a vector field \, \textbf{A} . If there exists a scalar field \, \Phi \, such that

\, A = \text{grad} \, \Phi \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \;\; (3.8)

the vector field \, \textbf{A} \, is said to have the (scalar) potential \, \Phi .

The potential for \, \textbf{A} \, is determined up to an arbitrary constant. Because if

\, A = \text{grad} \, {\Phi}_1 = \text{grad} \, {\Phi}_2 \,

holds true, then we have \, \text{grad} \, ( {\Phi}_1 - {\Phi}_2 ) = 0 \, and this implies that \, {\Phi}_1 - {\Phi}_2 = c , that is, \, {\Phi}_1 = {\Phi}_2 + c .

Theorem 3.5 If the continuously differentiable vector field \, \textbf{A} \, has a potential, then we have

\, \frac{\partial A_y}{\partial x} = \frac{\partial A_x}{\partial y} \, , \quad \small \text {cykl.} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \;\;\;\; (3.9)

Proof:

\, \frac{\partial A_y}{\partial x} = \frac{\partial}{\partial x} \frac{\partial \Phi}{\partial y} = \frac{\partial}{\partial y} \frac{\partial \Phi}{\partial x} = \frac{\partial A_x}{\partial y} \, , \quad \small \text {cykl.} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \boxdot

Reversely, we will see (in chapter 7) that from (3.9) and certain prerequisites one can conclude that the vector field \, \textbf{A} \, has a potential.

Often a potential \, U(\textbf{r}) \, for \, \textbf{A} is defined by the equation:

\, \textbf{A} = - \text{grad} \, U. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad (3.10)

The relationship between \, \Phi \, och \, U \, is given by

\, U(\textbf{r}) = - \Phi(\textbf{r}) + c. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \, (3.11)

/////// End of the translation from from Ramgard (1992).

///////

The complex exponential function

///////

Electromagnetic radiation

A planar electromagnetic wave:

The interactive simulation that created this movie.

The electric part of the wave: \, E(\mathbf{\hat{k}}, \mathbf{x}, \omega, t) \, = \, e^{ \, i \,(\mathbf{\hat{k}} \cdot \mathbf{x} \, - \, \omega \, t)} \,

The magnetic part of the wave: \, B(\mathbf{\hat{k}}, \mathbf{x}, \omega, t) \, = \, e^{ \, i \, (\mathbf{\hat{k}} \cdot \mathbf{x} \, - \, (\omega \, + \, \pi/2) \, t)} \,

The entire wave: \, E_m(\mathbf{\hat{k}}, \mathbf{x}, \omega, t) \, = \, E(\mathbf{\hat{k}}, \mathbf{x}, \omega, t) \, + \, B(\mathbf{\hat{k}}, \mathbf{x}, \omega, t) \,

Its Poynting vector : \, S \, = \, \frac{1}{{\mu}_0} \, E \, \times \, B

///////

Electromagnetism

Maxwell and Dirac theories as an already unified theory

Conceptual background:

Geometric Algebra

Clifford Algebra

Historical background:

The Evolution Of Geometric Arithmetic

///////

Divergence and curl: The language of Maxwell's equations, fluid flow, and more
(Steven Strogatz on YouTube):

///////

11 thoughts on “Vector Analysis

  1. You actually make it seem so easy with your presentation but I find this topic to be actually something that I think I would never understand. It seems too complex and very broad for me. I am looking forward to your next post, I will try to get the hang of it!

  2. Great goods from you, man. I have be mindful your stuff previous to and you’re simply extremely fantastic. I really like what you’ve acquired here, certainly like what you’re stating and the way during which you are saying it. You’re making it enjoyable and you still take care of to keep it sensible. I can not wait to read far more from you. That is really a terrific site.

  3. We are a group of volunteers and starting a brand new scheme in our community. Your site offered us with helpful info to work on. You’ve done an impressive task and our entire group will likely be grateful to you.

  4. Appreciating the commitment you put into your blog and detailed information you provide. It’s awesome to come across a blog every once in a while that isn’t the same unwanted rehashed information. Fantastic read! I’ve saved your site and I’m including your RSS feeds to my Google account.

  5. If you want to improve your experience simply keep visiting this web page and be updated with the most recent news update posted here.

  6. It is truly a great and useful piece of information. I am happy that you simply shared this helpful information with us. Please stay us up to date like this. Thanks for sharing.

  7. Hello There. I discovered your weblog the usage of msn. This is an extremely well written article. I will be sure to bookmark it and come back to learn extra of your helpful info. Thank you for the post. I will definitely comeback.

Leave a Reply