This section considers functions from $R^n$ into $R^m$ where one or both of $n$ or $m$ is greater than $1$:
* functions $f:R \rightarrow R^m$ are called univariate functions.
* functions $f:R^n \rightarrow R$ are called scalar-valued functions.
* function $f:R \rightarrow R$ are univariate, scalar-valued functions.
* functions $\vec{r}:R\rightarrow R^m$ are parameterized curves. The trace of a parameterized curve is a path.
* functions $F:R^n \rightarrow R^m$, may be called vector fields in applications. They are also used to describe transformations.
When $m>1$ a function is called *vector valued*.
When $n>1$ the argument may be given in terms of components, e.g. $f(x,y,z)$; with a point as an argument, $F(p)$; or with a vector as an argument, $F(\vec{a})$. The identification of a point with a vector is done frequently.
## Limits
Limits when $m > 1$ depend on the limits of each component existing.
Limits when $n > 1$ are more complicated. One characterization is a limit at a point $c$ exists if and only if for *every* continuous path going to $c$ the limit along the path for every component exists in the univariate sense.
## Derivatives
The derivative of a univariate function, $f$, at a point $c$ is defined by a limit:
and as a function by considering the mapping $c$ into $f'(c)$. A characterization is it is the value for which
$$
|f(c+h) - f(h) - f'(c)h| = \mathcal{o}(|h|),
$$
That is, after dividing the left-hand side by $|h|$ the expression goes to $0$ as $|h|\rightarrow 0$. This characterization will generalize with the norm replacing the absolute value, as needed.
### Parameterized curves
The derivative of a function $\vec{r}: R \rightarrow R^m$, $\vec{r}'(t)$, is found by taking the derivative of each component. (The function consisting of just one component is univariate.)
The derivative satisfies
$$
\| \vec{r}(t+h) - \vec{r}(t) - \vec{r}'(t) h \| = \mathcal{o}(|h|).
$$
The derivative is *tangent* to the curve and indicates the direction of travel.
The **tangent** vector is the unit vector in the direction of $\vec{r}'(t)$:
$$
\hat{T} = \frac{\vec{r}'(t)}{\|\vec{r}(t)\|}.
$$
The path is parameterized by *arc* length if $\|\vec{r}'(t)\| = 1$ for all $t$. In this case an "$s$" is used for the parameter, as a notational hint: $\hat{T} = d\vec{r}/ds$.
The **normal** vector is the unit vector in the direction of the derivative of the tangent vector:
$$
\hat{N} = \frac{\hat{T}'(t)}{\|\hat{T}'(t)\|}.
$$
In dimension $m=2$, if $\hat{T} = \langle a, b\rangle$ then $\hat{N} = \langle -b, a\rangle$ or $\langle b, -a\rangle$ and $\hat{N}'(t)$ is parallel to $\hat{T}$.
In dimension $m=3$, the **binormal** vector, $\hat{B}$, is the unit vector $\hat{T}\times\hat{N}$.
The [Frenet-Serret]() formulas define the **curvature**, $\kappa$, and the **torsion**, $\tau$, by
These formulas apply in dimension $m=2$ with $\hat{B}=\vec{0}$.
The curvature, $\kappa$, can be visualized by imagining a circle of radius $r=1/\kappa$ best approximating the path at a point. (A straight line would have a circle of infinite radius and curvature $0$.)
The chain rule says $(\vec{r}(g(t))' = \vec{r}'(g(t)) g'(t)$.
### Scalar functions
A scalar function, $f:R^n\rightarrow R$, $n > 1$ has a **partial derivative** defined. For $n=2$, these are:
The generalization to $n>2$ is clear - the partial derivative in $x_i$ is the derivative of $f$ when the *other* $x_j$ are held constant.
This may be viewed as the derivative of the univariate function $(f\circ\vec{r})(t)$ where $\vec{r}(t) = p + t \hat{e}_i$, $\hat{e}_i$ being the unit vector of all $0$s except a $1$ in the $i$th component.
The **gradient** of $f$, when the limits exist, is the vector-valued function for $R^n$ to $R^n$:
The gradient is viewed as a column vector. If the dot product above is viewed as matrix multiplication, then it would be written $\nabla{f}' \Delta{\vec{x}}$.
The **directional derivative** of $f$ in the direction $\vec{v}$ is $\vec{v}\cdot\nabla{f}$, which can be seen as the derivative of the univariate function $(f\circ\vec{r})(t)$ where $\vec{r}(t) = p + t \vec{v}$.
For the function $z=f(x,y)$ the gradient points in the direction of steepest ascent. Ascent is seen in the $3$d surface, the gradient is $2$ dimensional.
For a function $f(\vec{x})$, a **level curve** is the set of values for which $f(\vec{x})=c$, $c$ being some constant. Plotted, this may give a curve or surface (in $n=2$ or $n=3$). The gradient at a point $\vec{x}$ with $f(\vec{x})=c$ will be *orthogonal* to the level curve $f=c$.
Partial derivatives are scalar functions, so will themselves have partial derivatives when the limits are defined. The notation $f_{xy}$ stands for the partial derivative in $y$ of the partial derivative of $f$ in $x$. [Schwarz]()'s theorem says the order of partial derivatives will not matter (e.g., $f_{xy} = f_{yx}$) provided the higher-order derivatives are continuous.
The chain rule applied to $(f\circ\vec{r})(t)$ says:
This can be viewed as being comprised of row vectors, each being the individual gradients; or as column vectors each being the vector of partial derivatives for a given variable.
The **chain rule** for $F:R^n \rightarrow R^m$ composed with $G:R^k \rightarrow R^n$ is:
$$
d_{F\circ G}(a) = d_F(G(a)) d_G(a),
$$
That is the total derivative of $F$ at the point $G(a)$ times (matrix multiplication) the total derivative of $G$ at $a$. The dimensions work out as $d_F$ is $m\times n$ and $d_G$ is $n\times k$, so $d_(F\circ G)$ will be $m\times k$ and $F\circ{G}: R^k\rightarrow R^m$.
A scalar function $f:R^n \rightarrow R$ and a parameterized curve $\vec{r}:R\rightarrow R^n$ composes to yield a univariate function. The total derivative of $f\circ\vec{r}$ satisfies:
The curl measures the circulation in a vector field. In dimension $n=3$ it *points* in the direction of the normal of the plane of maximum circulation with direction given by the right-hand rule.
A vector field whose curl is identically of magnitude $0$ is called **irrotational**.
The $\nabla$ operator is the *formal* vector
$$
\nabla = \langle
\frac{\partial}{\partial{x}},
\frac{\partial}{\partial{y}},
\frac{\partial}{\partial{z}}
\rangle.
$$
The gradient is then scalar "multiplication" on the left: $\nabla{f}$.
The divergence is the dot product on the left: $\nabla\cdot{F}$.
The curl is the the cross product on the left: $\nabla\times{F}$.
These operations satisfy two vanishing properties:
* The curl of a gradient is the zero vector: $\nabla\times\nabla{f}=\vec{0}$
* The divergence of a curl is $0$: $\nabla\cdot(\nabla\times F)=0$
[Helmholtz]() decomposition theorem says a vector field ($n=3$) which vanishes rapidly enough can be expressed in terms of $F = -\nabla\phi + \nabla\times{A}$. The left term will be irrotational (no curl) and the right term will be incompressible (no divergence).
## Integrals
The definite integral, $\int_a^b f(x) dx$, for a bounded univariate function is defined in terms Riemann sums, $\lim \sum f(c_i)\Delta{x_i}$ as the maximum *partition* size goes to $0$. Similarly the integral of a bounded scalar function $f:R^n \rightarrow R$ over a box-like region $[a_1,b_1]\times[a_2,b_2]\times\cdots\times[a_n,b_n]$ can be defined in terms of a limit of Riemann sums. A Riemann integrable function is one for which the upper and lower Riemann sums agree in the limit. A characterization of a Riemann integrable function is that the set of discontinuities has measure $0$.
If $f$ and the partial functions ($x \rightarrow f(x,y)$ and $y \rightarrow f(x,y)$) are Riemann integrable, then Fubini's theorem allows the definite integral to be performed iteratively:
* As $f$ is bounded, let $m \leq f(x) \leq M$ for all $x$ in $R$. Then
$$
m V(R) \leq \iint_R f(x) dV \leq MV(R).
$$
* If $f$ and $g$ are integrable *and* $f(x) \leq g(x)$, then the integrals have the same property, namely $\iint_R f dV \leq \iint_R gdV$.
* If $S \subset R$, both closed rectangles, then if $f$ is integrable over $R$ it will be also over $S$ and, when $f\geq 0$, $\iint_S f dV \leq \iint_R fdV$.
* If $f$ is bounded and integrable, then $|\iint_R fdV| \leq \iint_R |f| dV$.
In two dimensions, we have the following interpretations:
To compute integrals over non-box-like regions, Fubini's theorem may be utilized. Alternatively, a **transformation** of variables
### Line integrals
For a parameterized curve, $\vec{r}(t)$, the **line integral** of a scalar function between $a \leq t \leq b$ is defined by: $\int_a^b f(\vec{r}(t)) \| \vec{r}'(t)\| dt$. For a path parameterized by arc-length, the integral is expressed by $\int_C f(\vec{r}(s)) ds$ or simply $\int_C f ds$, as the norm is $1$ and $C$ expresses the path.
A Jordan curve in two dimensions is a non-intersecting continuous loop in the plane. The Jordan curve theorem states that such a curve divides the plane into a bounded and unbounded region. The curve is *positively* parameterized if the the bounded region is kept on the left. A line integral over a Jordan curve is denoted $\oint_C f ds$.
Some interpretations: $\int_a^b \| \vec{r}'(t)\| dt$ computes the *arc-length*. If the path represents a wire with density $\rho(\vec{x})$ then $\int_a^b \rho(\vec{r}(t)) \|\vec{r}'(t)\| dt$ computes the mass of the wire.
The line integral is also defined for a vector field $F:R^n \rightarrow R^n$ through $\int_a^b F(\vec{r}(t)) \cdot \vec{r}'(t) dt$. When parameterized by arc length, this becomes $\int_C F(\vec{r}(s)) \cdot \hat{T} ds$ or more simply $\int_C F\cdot\hat{T}ds$. In dimension $n=2$ if $\hat{N}$ is the normal, then this line integral (the flow) is also of interest $\int_a^b F(\vec{r}(t)) \cdot \hat{N} dt$ (this is also expressed by $\int_C F\cdot\hat{N} ds$).
When $F$ is a *force field*, then the interpretation of $\int_a^b F(\vec{r}(t)) \cdot \vec{r}'(t) dt$ is the amount of *work* to move an object from $\vec{r}(a)$ to $\vec{r}(b)$. (Work measures force applied times distance moved.)
A **conservative force** is a force field within an open region $R$ with the property that the total work done in moving a particle between two points is independent of the path taken. (Similarly, integrals over Jordan curves are zero.)
The gradient theorem or **fundamental theorem of line integrals** states if $\phi$ is a scalar function then the vector field $\nabla{\phi}$ (if continuous in $R$) is a conservative field. That is if $q$ and $p$ are points, $C$ any curve in $R$, and $\vec{r}$ a parameterization of $C$ over $[a,b]$ that $\phi(p) - \phi(q) = \int_a^b \nabla{f}(\vec{r}(t)) \cdot \vec{r}'(t) dt$.
If $\phi$ is a scalar function producing a field $\nabla{\phi}$ then in dimensions $2$ and $3$ the curl of $\nabla{\phi}$ is zero when the functions involved are continuous. Conversely, if the curl of a force field, $F$, is zero *and* the derivatives are continuous in a *simply connected* domain, then there exists a scalar potential function, $\phi,$ with $F = -\nabla{\phi}$.
In dimension $2$, if $F$ describes a flow field, the integral $\int_C F \cdot\hat{N}ds$ is interpreted as the flow across the curve $C$; when $C$ is a closed curve $\oint_C F\cdot\hat{N}ds$ is interpreted as the flow out of the region, when $C$ is positively parameterized.
**Green's theorem** states if $C$ is a positively oriented Jordan curve in the plane bounding a region $D$ and $F$ is a vector field $F:R^2 \rightarrow R^2$ then $\oint_C F\cdot\hat{T}ds = \iint_D \text{curl}(F) dA$.
Green's theorem can be re-expressed in flow form: $\oint_C F\cdot\hat{N}ds=\iint_D\text{divergence}(F)dA$.
For $F=\langle -y,x\rangle$, Green's theorem says the area of $D$ is given by $(1/2)\oint_C F\cdot\vec{r}' dt$. Similarly, if $F=\langle 0,x\rangle$ or $F=\langle -y,0\rangle$ then the area is given by $\oint_C F\cdot\vec{r}'dt$. The above follows as $\text{curl}(F)$ is $2$ or $1$. Similar formulas can be given to compute the centroids, by identifying a vector field with $\text{curl}(F) = x$ or $y$.
### Surface integrals
A surface in $3$ dimensions can be described by a scalar function $z=f(x,y)$, a parameterization $F:R^2 \rightarrow R^3$ or as a level curve of a scalar function $f(x,y,z)$. The second case, covers the first through the parameterization $(x,y) \rightarrow (x,y,f(x,y)$. For a parameterization of a surface, $\Phi(u,v) = \langle \Phi_x, \Phi_y, \Phi_z\rangle$, let $\partial{\Phi}/\partial{u}$ be the $3$-d vector $\langle \partial{\Phi_x}/\partial{u}, \partial{\Phi_y}/\partial{u}, \partial{\Phi_z}/\partial{u}\rangle$, similarly define $\partial{\Phi}/\partial{v}$. As vectors, these lie in the tangent plane to the surface and this plane has normal vector $\vec{N}=\partial{\Phi}/\partial{u}\times\partial{\Phi}/\partial{v}$. For a closed surface, the parametrization is positive if $\vec{N}$ is an outward pointing normal. Let the *surface element* be defined by $\|\vec{N}\|$.
The surface integral of a scalar function $f:R^3 \rightarrow R$ for a parameterization $\Phi:R \rightarrow S$ is defined by
If $F$ is a vector field, the surface integral may be defined as a flow across the boundary through
$$
\iint_R F(\Phi(u,v)) \cdot \vec{N} du dv =
\iint_R (F \cdot \hat{N}) \|\frac{\partial{\Phi}}{\partial{u}} \times \frac{\partial{\Phi}}{\partial{v}}\| du dv = \iint_S (F\cdot\hat{N})dS
$$
### Stokes' theorem, divergence theorem
**Stokes' theorem** states that in dimension $3$ if $S$ is a smooth surface with boundary $C$ – *oriented* so the right-hand rule gives the choice of normal for $S$ – and $F$ is a vector field with continuous partial derivatives then:
$$
\iint_S (\nabla\times{F}) \cdot \hat{N} dS = \oint_C F ds.
$$
Stokes' theorem has the same formulation as Green's theorem in dimension $2$, where the surface integral is just the $2$-dimensional integral.
Stokes' theorem is used to show a vector field $F$ with zero curl is conservative if $F$ is continuous in a simply connected region.
Stokes' theorem is used in Physics, for example, to relate the differential and integral forms of $2$ of Maxwell's equations.
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The **divergence theorem** states if $V$ is a compact volume in $R^3$ with piecewise smooth boundary $S=\partial{V}$ and $F$ is a vector field with continuous partial derivatives then:
