Line and surface integrals

Many physical quantities are vectors that vary from point to point: the velocity of water in a river, the wind over a building, the electric field around a cable, the force of gravity. These are vector fields. Engineers constantly need two kinds of totals involving them: how much a field pushes along a path (the work done moving a load, the circulation of a fluid around a wing) and how much of it flows through a surface (water through a screen, heat through a wall, electric flux through a closed surface). Line integrals and surface integrals measure exactly these.

This chapter introduces vector fields, defines line integrals of scalar functions and of vector fields, develops the special and very useful case of conservative fields with potential functions, and then moves to surfaces: describing them with parameters, finding their area, and computing the flux of a vector field through them. These are the ingredients of the great theorems in Lesson 5.

Vector fields

A vector field assigns a vector to each point: in 2D, \(\mathbf{F}(x, y) = P(x, y)\,\mathbf{i} + Q(x, y)\,\mathbf{j}\); in 3D, a third component \(R\,\mathbf{k}\) is added. A plot shows arrows at a grid of points.

Some fields engineers meet constantly:

• Velocity fields in fluids: \(\mathbf{v}(x, y, z)\), the velocity of the fluid particle at each point.
• Force fields: gravity near the Earth is \(\mathbf{F} = -mg\,\mathbf{k}\), constant; far from the Earth it is \(-GMm\,\mathbf{r}/|\mathbf{r}|^3\), pointing toward the centre.
• Gradient fields: \(\nabla f\) for any scalar function, such as the heat flux \(-k\nabla T\) and the electric field \(-\nabla V\).
• The rotation field \(\mathbf{F} = (-y, x)\), the velocity of a rigid disk spinning at 1 rad/s.

Line integrals of scalar functions

To add up a scalar quantity along a curve C, such as the mass of a curved wire with varying density, parametrize the curve as \(\mathbf{r}(t) = (x(t), y(t))\) for \(a \le t \le b\). Each small piece has length \(ds = |\mathbf{r}'(t)|\,dt\), so

Line integrals of vector fields: work

If a force field \(\mathbf{F}\) acts on an object moving along C, the work done is the sum of the force component along the path times each small displacement:

Only the part of the force tangent to the path does work; the perpendicular part does none. Reversing the direction of travel changes the sign. When C is a closed loop, the integral is written \(\oint_C\) and called the circulation, a key quantity in fluid mechanics: the lift on an aircraft wing is proportional to the circulation of air around it.

Conservative fields and potential functions

For some fields, the work between two points is the same along every path. These conservative fields are exactly the gradient fields: \(\mathbf{F} = \nabla\phi\) for some scalar potential function \(\phi\). For them, the fundamental theorem for line integrals reduces any line integral to a subtraction:

Consequences: the work depends only on the endpoints, and the circulation around any closed loop is zero. Gravity and the electrostatic force are conservative, which is why potential energy (\(mgh\), \(qV\)) makes sense: the work to lift a load to a given height is the same whichever route the crane takes. Friction and the rotation field above are not conservative.

Testing for a conservative field

In 2D, on a region without holes, \(\mathbf{F} = (P, Q)\) is conservative exactly when

This follows from Clairaut's theorem: if \(P = \phi_x\) and \(Q = \phi_y\), both sides equal \(\phi_{xy}\). In 3D the test is that the curl (Lesson 5) is zero.

Parametric surfaces

Just as curves are described by one parameter, surfaces are described by two: \(\mathbf{r}(u, v) = (x(u, v), y(u, v), z(u, v))\). Standard examples:

• The graph \(z = g(x, y)\): \(\mathbf{r}(x, y) = (x, y, g(x, y))\).
• A cylinder of radius R: \(\mathbf{r}(\theta, z) = (R\cos\theta, R\sin\theta, z)\).
• A sphere of radius R: \(\mathbf{r}(\phi, \theta) = (R\sin\phi\cos\theta, R\sin\phi\sin\theta, R\cos\phi)\).

The partial derivatives \(\mathbf{r}_u\) and \(\mathbf{r}_v\) are tangent to the surface, so their cross product \(\mathbf{r}_u \times \mathbf{r}_v\) is normal to it, and its length is the area scale factor. A small patch has area

For a graph \(z = g(x, y)\), this becomes \(dS = \sqrt{1 + g_x^2 + g_y^2}\,dA\).

Flux: flow through a surface

The flux of a vector field \(\mathbf{F}\) through a surface S measures how much of the field passes through it. Choose a unit normal \(\mathbf{n}\) on S (this choice is the orientation; for closed surfaces, outward is standard). Only the normal component of F carries anything through the surface, so

If \(\mathbf{F}\) is a fluid velocity, the flux is the volume flow rate through the surface (m³/s). If \(\mathbf{F}\) is a heat flux vector (W/m²), the flux is the rate of heat flow (W). If \(\mathbf{F}\) is an electric field, the flux appears in Gauss's law.