7 Vector Calculus

Read Chapter 16 in Stewart.

7.1 Vector fields

Definition 7.1 Let \(D\) be a domain on \(\mathbb{R}^n\). A vector field on \(\mathbb{R}^n\) is a function \(\mathbf{F}: D \to \mathbb{R}^n\) that assign each point \(\mathbf{x}\in D\) to a vector \(\mathbf{F}(\mathbf{x}) \in \mathbb{R}^n\).

In \(\mathbb{R}^2\), one typically write the vector fields in terms of component functions \(P, Q\) \[\mathbf{F}(x,y) = P(x,y) \mathbf{i} + Q(x,y) \mathbf{j}\,.\]

In \(\mathbb{R}^3\), one typically write the vector fields in terms of component functions \(P, Q, R\) \[\mathbf{F}(x,y,z) = P(x,y) \mathbf{i} + Q(x,y) \mathbf{j} + R(x,y,z) \mathbf{k}\,.\]

Example 7.1 Newton’s Law of Gravitation \[\begin{equation*} \mathbf{F}(\mathbf{x}) = - \frac{m M G}{| \mathbf{x}|^3 } \mathbf{x} \,, \end{equation*}\] where \(\mathbf{x}\) is the position in \(\mathbb{R}^3\).

Example 7.2 Coulomb’s Law for the electric force exerted by an electric charge \(Q\) at the origin on another charge \(q\) at a point \(\mathbf{x}\in \mathbb{R}^3\). \[\begin{equation*} \mathbf{F}(\mathbf{x}) = \frac{ \epsilon q Q}{|\mathbf{x}|^3} \mathbf{x} \,. \end{equation*}\]

7.2 Line integrals

Let’s focus on \(\mathbb{R}^2\). We now perform a Riemann-sum-like action.

Definition 7.2 Let \(C\) be a curve. The line integral of \(f\) along \(C\) is defined as \[\begin{equation*} \int_C f(x,y) \, ds = \lim_{n\to \infty} \sum_{i=1}^n f(x_i^*, y_i^*) \Delta s_i \,, \end{equation*}\] where \(\Delta s_i\) is the length of a subarc of \(C\).

Proposition 7.1 Suppose \(C\) is smooth and is parametrized by \(\mathbf{r}(t), a\leq t \leq b\). Then \[\begin{equation*} \int_C f(x,y) \, ds = \int_a^b f(\mathbf{r}(t)) |\mathbf{r}'(t)| \, dt \,. \end{equation*}\]

Note: when integrating with respect to arc length like this, reverse the direction of traversing the curve \(C\) will NOT result in a change of sign of the final solution.

\[\begin{equation*} \int_{-C} f(x,y) \, ds = \int_C f(x,y) \, ds \,. \end{equation*}\]

Now we define line integrals of vector fields.

Definition 7.3 Let \(\mathbf{F}\) be a continuous vector field defined on a curve \(C\). Then the line integral of \(\mathbf{F}\) along \(C\) is defined as \[\begin{equation*} \int_C \mathbf{F} \cdot d \mathbf{r} = \int_C \mathbf{F}\cdot \mathbf{T} \, ds \,, \end{equation*}\] where \(\mathbf{T}\) is the unit tangent vector.

Proposition 7.2 Suppose \(C\) is smooth and parametrized by \(\mathbf{r}(t), a \leq t \leq b\). Then \[\begin{equation*} \int_C \mathbf{F} \cdot d \mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t) \, dt \end{equation*}\]

We also use the following notations

\[\begin{align*} \int_C f(x,y) dx := \int_a^b f(x(t), y(t) ) \, x'(t) \, dt \,, \\ \int_C f(x,y) dy := \int_a^b f(x(t), y(t) ) \, y'(t) \, dt \,, \\ \end{align*}\]

We can abbreviate the above by \[\begin{equation*} \int_C P(x,y) dx + \int_C Q(x,y) dy = \int_C P(x,y) \, dx + Q(x,y) \, dy \,. \end{equation*}\]

So, \[\begin{equation*} \int_C \mathbf{F} \cdot d \mathbf{r} = \int_C P \, dx + Q \, dy \,. \end{equation*}\]

Note: as oppose to integrating the arc length, reversing the order of the above integrals will change the sign of the integral. This is because the arc length is always positive, while \(\Delta x\) and \(\Delta y\) could be either positive or negative.

\[\begin{equation*} \int_{-C} P(x,y) \, dx + Q(x,y) \, dy = -\int_C P(x,y) \, dx + Q(x,y) \, dy \,. \end{equation*}\]

Theorem 7.1 (Fundamental Theorem for line integrals) Let \(C\) be a smooth curve given by the parametrization \(\mathbf{r}(t)\), \(a \leq t \leq b\). Let \(f\) be a differentiable function of two or three variables whose gradient vector \(\nabla f\) is continuous on \(C\). Then, \[\begin{equation*} \int_C \nabla f \cdot d\mathbf{r} = f(\mathbf{r}(b)) - f(\mathbf{r}(a)) \,. \end{equation*}\]

Definition 7.4 A closed curve is a curve that starts and ends at the same point. A simple closed curve is a closed curve that never crosses itself.

Sometimes, if \(C\) is a closed curve, we signify it by the following notation \[\begin{equation*} \oint_C \nabla f \cdot d\mathbf{r} \,. \end{equation*}\]

Corollary 7.1 If \(C\) is a closed curve and \(f:\mathbb{R}^n \to \mathbb{R}\) is a smooth function, then \[\begin{equation*} \oint_C \nabla f \cdot d\mathbf{r} = 0 \,. \end{equation*}\]

Definition 7.5 A vector field \(\mathbf{F}\) is called a conservative vector field if it is the gradient of some scalar function, that is there exists a function \(f\) such that \[\begin{equation*} \nabla f = \mathbf{F} \,. \end{equation*}\]

Therefore, if \(\mathbf{F}\) is a conservative vector field, then \[\begin{equation*} \oint_C \mathbf{F} \cdot d\mathbf{r} = 0 \,. \end{equation*}\]

7.2.1 Independence of path

Suppose \(C_1\) and \(C_2\) are two piecewise smooth curves that have the same initial point \(A\) and end point \(B\). Then, \[\begin{equation*} \int_{C_1} \mathbf{F}\cdot d\mathbf{r} = \int_{C_2} \mathbf{F} \cdot d\mathbf{r} \end{equation*}\] whenever \(\mathbf{F}\) is conservative (Why?). The question is when is the converse true?

The following example is an example when the converse is not always true.

Example 7.3 Evaluate \[\begin{equation*} \int_{C_i} x^2 \, dy \,, \qquad i = 1,2 \end{equation*}\] where \(C_1\) is the line segments from \((-1,-1) \to (-1,1) \to (1,1)\) and \(C_2\) is the line segments from \((-1,-1) \to (1,-1) \to (1,1)\).

To further the discussion, we need a few definitions.

Definition 7.6 Let \(\mathbf{F}\) be a continuous vector field with domain \(D\), we say that the line integral \[\begin{equation*} \int_C \mathbf{F} \cdot d\mathbf{r} \end{equation*}\] is independent of path if \[\begin{equation*} \int_{C_1} \mathbf{F}\cdot d\mathbf{r} = \int_{C_2} \mathbf{F} \cdot d\mathbf{r} \end{equation*}\] for all paths that have the same starting and ending points.

Theorem 7.2 \(\int_C \mathbf{F}\cdot d\mathbf{r}\) is independent of path in \(D\) if and only if \(\oint_\Gamma \mathbf{F} \cdot d\mathbf{r} = 0\) for every closed path \(\Gamma\) in \(D\).

Definition 7.7 A domain \(D\) is said to be open if around each point, we can draw an open ball around it. A domain \(D\) is said to be connected if for any two points, there is a path that connect them together. A domain \(D\) is said to be simply connected if is connected and there’s no hole in it.

Theorem 7.3 Suppose \(\mathbf{F}\) is a vector field that is continuous on an open connected region \(D\). If \(\int_C \mathbf{F} \cdot d \mathbf{r}\) is independent of path in \(D\), then \(\mathbf{F}\) is a conservative vector field on \(D\).

Proof. todo

The above theorem gives a way to determine if a vector field is conservative or not, from the point of view of path independence. However, it is often difficult to check the path independence property as one has to integrate over ALL possible curves, and there are a lot of them…

Another way is to take inspiration from Clairaut’s theorem. The question is to determine whether \(\mathbf{F}\) is conservative, given the mixed partial derivatives of \(P\) and \(Q\) are the same, i.e., \[\begin{equation*} \frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}\,. \end{equation*}\] (Compare this with Clairaut’s)

Theorem 7.4 Let \(\mathbf{F} = P\mathbf{i} + Q\mathbf{j}\) be a vector field on an open simply connected region \(D\). Suppose that \(P\) and \(Q\) have continuous first-order partial derivatives and \[\begin{equation*} \frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x} \end{equation*}\] through out \(D\). Then \(\mathbf{F}\) is conservative.

Remark. The connectedness of \(D\) is crucial (why?).

7.3 Green’s Theorem

Theorem 7.5 (Green's Theorem) Let \(D\) be an open bounded simply connected domain in \(\mathbb{R}^2\), \(\Gamma\) be the boundary of \(D\), and \(\mathbf{F} = P\mathbf{i} + Q \mathbf{j}\) be a vector field. If \(P\) and \(Q\) have continuous partial derivatives on an open region that contains \(D\), then \[\begin{equation*} \int_\Gamma \mathbf{F} \cdot d \ell = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dA \,. \end{equation*}\]

7.4 Curl and Divergence

Definition 7.8 Let \(\mathbf{F}\) be a vector field in \(\mathbb{R}^3\). If all partial derivatives of \(P,Q,R\) exist, then we define \[\begin{equation*} \mathrm{curl}\,\mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k} \,. \end{equation*}\]

A different notation for \(\mathrm{curl} \, \mathbf{F}\) is \[\begin{equation*} \nabla \times \mathbf{F} \,. \end{equation*}\]

Theorem 7.6 If \(f\) is a function of 3 variables that has continuous second partial derivatives, then \[\begin{equation*} \nabla \times ( \nabla f) = 0 \,. \end{equation*}\]

Theorem 7.7 Suppose \(\mathbf{F}\) is a vector field on and simply connected domain \(D\) so that \(P,Q,R\) all have continuous partial derivatives. \(F\) is a conservative vector field if and only if \(\nabla \times \mathbf{F} = 0\).

Definition 7.9 Let \(\mathbf{F}\) be a vector field in \(\mathbb{R}^3\). If all partial derivatives of \(P,Q,R\) exist, then we define \[\begin{equation*} \mathrm{div}\,\mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \,. \end{equation*}\]

A different notation for \(\mathrm{div} \, \mathbf{F}\) is \[\begin{equation*} \nabla \cdot \mathbf{F} \,. \end{equation*}\]

Theorem 7.8 Suppose \(\mathbf{F}\) is a vector field on a domain \(D\) and \(P,Q,R\) have continuous second-order partial derivatives. Then, \[\begin{equation*} \nabla \cdot (\nabla \times \mathbf{F}) = 0 \,. \end{equation*}\]

7.5 Surface integrals

7.5.1 Parametric surfaces

Similar to the way we parametrize a curve by a one-variable vector function \(\mathbf{r}(t)\), we can parametrize a surface by a two-variable vector function \(\mathbf{r}(u,v)\).

We will only deal with surfaces in \(\mathbb{R}^3\) in this section. So, the parametrization of a surface \(S\) should be \[\begin{equation*} \mathbf{r}: D\subseteq \mathbb{R}^2 \to \mathbb{R}^3 \,. \end{equation*}\] We often write \[\begin{equation*} \mathbf{r}(u,v) = x(u,v) \mathbf{i} + y(u,v) \mathbf{j} + z(u,v) \mathbf{k} \,. \end{equation*}\]

From this parametrization, we get to talk about the tangent plane of \(S\) at the point \(\mathbf{r}(u,v)\), which is the plane that contains two tangent vectors \[\begin{equation*} \mathbf{r}_u (u,v) = \frac{\partial x}{\partial u} \mathbf{i} + \frac{\partial y}{\partial u} \mathbf{j} + \frac{\partial z}{\partial u} \mathbf{k} \,, \end{equation*}\] and \[\begin{equation*} \mathbf{r}_v (u,v) = \frac{\partial x}{\partial v} \mathbf{i} + \frac{\partial y}{\partial v} \mathbf{j} + \frac{\partial z}{\partial v} \mathbf{k} \,. \end{equation*}\]

7.5.2 Surface integral

Definition 7.10 Let \(S\) be a surface with parametrization. The surface integral of \(f\) over the surface \(S\) is \[\begin{equation*} \iint_S f(x,y,z) \, dS = \lim_{m,n\to \infty} \sum_{i=1}^m \sum_{j=1}^n f(P_{ij}^*) \Delta S_{ij} \,. \end{equation*}\]

Similarly to the line integral, one can show that \[\begin{equation*} \iint_S f(x,y,z) \, dS = \iint_D f(\mathbf{r}(u,v)) | \mathbf{r_u}\times \mathbf{r_v} | \, dA \,. \end{equation*}\]

7.5.3 Orientation of the surface

Given a surface \(S\), we define the orientation of it as following

1. If \(S\) has a boundary, then the positive orientation of the surface is that when one walks along the boundary of the surface with the head points in that direction, the surface is on the left.

2. If \(S\) does not have a boundary, then the positive orientation is the direction of the outward normal vector.

7.5.4 Surface integral of vector fields

Definition 7.11 If \(\mathbf{F}\) is a continuous vector field on an oriented surface \(S\) (parametrized by \(\mathbf{r}(u,v)\)) with unit normal vector \(\mathbf{n}\), then the surface integral of \(\mathbf{F}\) over \(S\) is \[\begin{equation*} \iint_S \mathbf{F}\cdot \, d\mathbf{S} = \iint_S \mathbf{F}\cdot \mathbf{n} \, dS = \iint_S \mathbf{F}\cdot (\mathbf{r}_u\times \mathbf{r}_v) \, dA \,. \end{equation*}\] The integral is called the flux of \(\mathbf{F}\) across \(S\).

7.6 Stokes’ and Divergence Theorem

Theorem 7.9 (Stokes' Theorem) Let \(S\) be an oriented smooth surface that is bounded by a simple closed smooth boundary curve \(\partial S\) with positive orientation. Let \(\mathbf{F}\) be a vector field whose components have continuous partial derivatives on an open region in \(\mathbb{R}^3\) that contains \(S\). Then \[\begin{equation*} \int_{\partial S} \mathbf{F} \cdot \, d\mathbf{r} = \iint_S \nabla \times \mathbf{F} \cdot d\mathbf{S} \,. \end{equation*}\]

The boundary of an area is a curve. Similarly, the boundary of a solid is a surface.

Theorem 7.10 (Divergence Theorem) Let \(E\) be a simple solid region and let surface \(\partial E\) be the boundary of \(E\), given with positive (outward) orientation. Let \(\mathbf{F}\) be a vector field whose components have continuous partial derivatives. Then, \[\begin{equation*} \iint_{\partial E} \mathbf{F} \cdot d\mathbf{S} = \iiint_E \mathrm{div} \mathbf{F} \, dV \,. \end{equation*}\]