63  Quick Review of Vector Calculus

This section considers functions from \(R^n\) into \(R^m\) where one or both of \(n\) or \(m\) is greater than \(1\):

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.

63.1 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.

63.2 Derivatives

The derivative of a univariate function, \(f\), at a point \(c\) is defined by a limit:

\[ f'(c) = \lim_{h\rightarrow 0} \frac{f(c+h)-f(c)}{h}, \]

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.

63.2.1 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

\[\begin{align*} \frac{d\hat{T}}{ds} &= & \kappa \hat{N} &\\ \frac{d\hat{N}}{ds} &= -\kappa\hat{T} & & + \tau\hat{B}\\ \frac{d\hat{B}}{ds} &= & -\tau\hat{N}& \end{align*}\]

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 \((\vec{r}(g(t))' = \vec{r}'(g(t)) g'(t)\).

63.2.2 Scalar functions

A scalar function, \(f:R^n\rightarrow R\), \(n > 1\) has a partial derivative defined. For \(n=2\), these are:

\[\begin{align*} \frac{\partial{f}}{\partial{x}}(x,y) &= \lim_{h\rightarrow 0} \frac{f(x+h,y)-f(x,y)}{h}\\ \frac{\partial{f}}{\partial{y}}(x,y) &= \lim_{h\rightarrow 0} \frac{f(x,y+h)-f(x,y)}{h}. \end{align*}\]

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 \(R^n\):

\[ \nabla{f} = \langle \frac{\partial{f}}{\partial{x_1}}, \frac{\partial{f}}{\partial{x_2}}, \dots \frac{\partial{f}}{\partial{x_n}} \rangle. \]

The gradient satisfies:

\[ \|f(\vec{x}+\Delta{\vec{x}}) - f(\vec{x}) - \nabla{f}\cdot\Delta{\vec{x}}\| = \mathcal{o}(\|\Delta{\vec{x}\|}). \]

The gradient is viewed as a column vector. If the dot product above is viewed as matrix multiplication, then it would be written \(\nabla{f}' \Delta{\vec{x}}\).

Linearization is the approximation

\[ f(\vec{x}+\Delta{\vec{x}}) \approx f(\vec{x}) + \nabla{f}\cdot\Delta{\vec{x}}. \]

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:

\[ \frac{d(f\circ\vec{r})}{dt} = \nabla{f}(\vec{r}) \cdot \vec{r}'. \]

63.2.3 Vector-valued functions

For a function \(F:R^n \rightarrow R^m\), the total derivative of \(F\) is the linear operator \(d_F\) satisfying:

\[ \|F(\vec{x} + \vec{h})-F(\vec{x}) - d_F \vec{h}\| = \mathcal{o}(\|\vec{h}\|) \]

For \(F=\langle f_1, f_2, \dots, f_m\rangle\) the total derivative is the Jacobian, a \(m \times n\) matrix of partial derivatives:

\[ J_f = \begin{bmatrix} \frac{\partial f_1}{\partial x_1} &\quad \frac{\partial f_1}{\partial x_2} &\dots&\quad\frac{\partial f_1}{\partial x_n}\\ \frac{\partial f_2}{\partial x_1} &\quad \frac{\partial f_2}{\partial x_2} &\dots&\quad\frac{\partial f_2}{\partial x_n}\\ &&\vdots&\\ \frac{\partial f_m}{\partial x_1} &\quad \frac{\partial f_m}{\partial x_2} &\dots&\quad\frac{\partial f_m}{\partial x_n} \end{bmatrix}. \]

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:

\[ d_f(\vec{r}) d\vec{r} = \nabla{f}(\vec{r}(t))' \vec{r}'(t) = \nabla{f}(\vec{r}(t)) \cdot \vec{r}'(t), \]

as above. (There is an identification of a \(1\times 1\) matrix with a scalar in re-expressing as a dot product.)

63.2.4 The divergence, curl, and their vanishing properties

Define the divergence of a vector-valued function \(F:R^n \rightarrow R^n\) by:

\[ \text{divergence}(F) = \frac{\partial{F_{x_1}}}{\partial{x_1}} + \frac{\partial{F_{x_2}}}{\partial{x_2}} + \cdots + \frac{\partial{F_{x_n}}}{\partial{x_n}}. \]

The divergence is a scalar function. For a vector field \(F\), it measures the microscopic flow out of a region.

A vector field whose divergence is identically \(0\) is called incompressible.

Define the curl of a two-dimensional vector field, \(F:R^2 \rightarrow R^2\), by:

\[ \text{curl}(F) = \frac{\partial{F_y}}{\partial{x}} - \frac{\partial{F_x}}{\partial{y}}. \]

The curl for \(n=2\) is a scalar function.

For \(n=3\) define the curl of \(F:R^3 \rightarrow R^3\) to be the vector field:

\[ \text{curl}(F) = \langle \ \frac{\partial{F_z}}{\partial{y}} - \frac{\partial{F_y}}{\partial{z}}, \frac{\partial{F_x}}{\partial{z}} - \frac{\partial{F_z}}{\partial{x}}, \frac{\partial{F_y}}{\partial{x}} - \frac{\partial{F_x}}{\partial{y}} \rangle. \]

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).

63.3 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:

\[ \iint_{R\times S}fdV = \int_R \left(\int_S f(x,y) dy\right) dx = \int_S \left(\int_R f(x,y) dx\right) dy. \]

The integral satisfies linearity and monotonicity properties that follow from the definitions:

  • For integrable \(f\) and \(g\) and constants \(a\) and \(b\):

\[ \iint_R (af(x) + bg(x))dV = a\iint_R f(x)dV + b\iint_R g(x) dV. \]

  • If \(R\) and \(R'\) are disjoint rectangular regions (possibly sharing a boundary), then the integral over the union is defined by linearity:

\[ \iint_{R \cup R'} f(x) dV = \iint_R f(x)dV + \iint_{R'} f(x) dV. \]

  • 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:

\[\begin{align*} \iint_R dA &= \text{area of } R\\ \iint_R \rho dA &= \text{mass with constant density }\rho\\ \iint_R \rho(x,y) dA &= \text{mass of region with density }\rho\\ \frac{1}{\text{area}}\iint_R x \rho(x,y)dA &= \text{centroid of region in } x \text{ direction}\\ \frac{1}{\text{area}}\iint_R y \rho(x,y)dA &= \text{centroid of region in } y \text{ direction} \end{align*}\]

In three dimensions, we have the following interpretations:

\[\begin{align*} \iint_VdV &= \text{volume of } V\\ \iint_V \rho dV &= \text{mass with constant density }\rho\\ \iint_V \rho(x,y) dV &= \text{mass of volume with density }\rho\\ \frac{1}{\text{volume}}\iint_V x \rho(x,y)dV &= \text{centroid of volume in } x \text{ direction}\\ \frac{1}{\text{volume}}\iint_V y \rho(x,y)dV &= \text{centroid of volume in } y \text{ direction}\\ \frac{1}{\text{volume}}\iint_V z \rho(x,y)dV &= \text{centroid of volume in } z \text{ direction} \end{align*}\]

To compute integrals over non-box-like regions, Fubini’s theorem may be utilized. Alternatively, a transformation of variables

63.3.1 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 \(y\).

63.3.2 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

\[ \iint_R f(\Phi(u,v)) \|\frac{\partial{\Phi}}{\partial{u}} \times \frac{\partial{\Phi}}{\partial{v}}\| du dv \]

If \(F\) is a vector field, the surface integral may be defined as a flow across the boundary through

\[ \iint_R F(\Phi(u,v)) \cdot \vec{N} du dv = \iint_R (F \cdot \hat{N}) \|\frac{\partial{\Phi}}{\partial{u}} \times \frac{\partial{\Phi}}{\partial{v}}\| du dv = \iint_S (F\cdot\hat{N})dS \]

63.3.3 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.

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:

\[ \iint_V (\nabla\cdot{F})dV = \oint_S (F\cdot\hat{N})dS. \]

The divergence theorem is available for other dimensions. In the \(n=2\) case, it is the alternate (flow) form of Green’s theorem.

The divergence theorem is used in Physics to express physical laws in either integral or differential form.