# Vector Analysis

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

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Related KMR-pages:

In Swedish:

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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 $\, d \textbf{r} \,$ in (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

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

Proof:

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

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).

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The complex exponential function

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A planar electro-magnetic wave:

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 : $\, S \, = \, \frac{1}{{\mu}_0} \, E \, \times \, B$

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Conceptual background:

Historical background:

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Divergence and curl: The language of Maxwell’s equations, fluid flow, and more

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