$$ \def\ab{\boldsymbol{a}} \def\bb{\boldsymbol{b}} \def\cb{\boldsymbol{c}} \def\db{\boldsymbol{d}} \def\eb{\boldsymbol{e}} \def\fb{\boldsymbol{f}} \def\gb{\boldsymbol{g}} \def\hb{\boldsymbol{h}} \def\jb{\boldsymbol{j}} \def\kb{\boldsymbol{k}} \def\nb{\boldsymbol{n}} \def\tb{\boldsymbol{t}} \def\ub{\boldsymbol{u}} \def\vb{\boldsymbol{v}} \def\xb{\boldsymbol{x}} \def\yb{\boldsymbol{y}} \def\Ab{\boldsymbol{A}} \def\Bb{\boldsymbol{B}} \def\Cb{\boldsymbol{C}} \def\Eb{\boldsymbol{E}} \def\Fb{\boldsymbol{F}} \def\Jb{\boldsymbol{J}} \def\Lb{\boldsymbol{L}} \def\Rb{\boldsymbol{R}} \def\Ub{\boldsymbol{U}} \def\xib{\boldsymbol{\xi}} \def\evx{\boldsymbol{e}_x} \def\evy{\boldsymbol{e}_y} \def\evz{\boldsymbol{e}_z} \def\evr{\boldsymbol{e}_r} \def\evt{\boldsymbol{e}_\theta} \def\evp{\boldsymbol{e}_r} \def\evf{\boldsymbol{e}_\phi} \def\evb{\boldsymbol{e}_\parallel} \def\omb{\boldsymbol{\omega}} \def\dA{\;d\Ab} \def\dS{\;d\boldsymbol{S}} \def\dV{\;dV} \def\dl{\mathrm{d}\boldsymbol{l}} \def\bfzero{\boldsymbol{0}} \def\Rey{\mathrm{Re}} \def\Real{\mathbb{R}} \def\grad{\boldsymbol\nabla} \newcommand{\dds}[1]{\frac{d{#1}}{ds}} \newcommand{\ddy}[2]{\frac{\partial{#1}}{\partial{#2}}} \newcommand{\ddt}[1]{\frac{d{#1}}{dt}} \newcommand{\DDt}[1]{\frac{\mathrm{D}{#1}}{\mathrm{D}t}} $$
2.1 Surfaces
A 2-d surface embedded in \(\Real^3\) can be defined parametrically as a map \(\xb(u,v)\) from \(\Real^2\to\Real^3\), for \(u\in[u_0,u_1]\) and \(v\in[v_0,v_1]\).
For each value of \(v\), the position \(\xb(u,v)\) describes a curve on the surface with tangent vector \(\displaystyle\ddy{\xb}{u}\).
Similarly, for each value of \(u\), we get a curve with tangent vector \(\displaystyle\ddy{\xb}{v}\).
If \((u,v)\) is a well-defined parametrisation, then their tangent vectors cannot be collinear. So the cross product of these tangent vectors allows us to define the unit normal to the surface: \[ \hat{\nb} = \frac{\displaystyle\ddy{\xb}{u}\times\ddy{\xb}{v}}{\displaystyle\left|\ddy{\xb}{u}\times\ddy{\xb}{v}\right|}. \]
The sign of \(\hat{\nb}\) depends on the ordering of \((u,v)\). A surface together with a specified (consistent) choice of \(\hat{\nb}\) is called an oriented surface.
Not all surfaces are orientable, meaning that a consistent choice of \(\hat{\nb}\) exists. An example of a non-orientable surface would be a Möbius strip.
A surface is called smooth if \(\hat{\nb}\) varies continuously over the surface (e.g. a sphere). It is called piecewise smooth if it can be divided into finitely many portions each of which is smooth (e.g. a cube).
A surface can be either open if it is bounded by a curve (e.g. the cone, or a hemisphere), or closed if it has no boundary (e.g. the sphere or torus). The boundary of an open surface \(S\) is a curve (or union of curves), often written \(\partial S\).
A simple type of surface is an explicit surface given by the height of a two-dimensional function, \(z=h(x,y)\).
2.2 Surface integrals of scalar fields
If \(f(\xb)=f(x,y,z)\) is a scalar field and \(S\) is a surface with parametrisation \(\xb(u,v)\), then we define the surface integral of \(f\) over \(S\) to be \[ \int_S f\,dS = \int_Uf(\xb(u,v))\left|\ddy{\xb}{u}\times\ddy{\xb}{v}\right|\,du\,dv, \] where \(U\) is the preimage of \(S\) in the \((u,v)\) plane.
To motivate this formula, note that the area element \(\displaystyle\left|\ddy{\xb}{u}\times\ddy{\xb}{v}\right|\,du\,dv\) is the area on the surface in \(\Real^3\) corresponding to an infinitesimal rectangle with sides \((du, dv)\) in the \((u,v)\) plane.
Proposition 2.1 If \((u,v)\) are (two of a set of) orthogonal curvilinear coordinates, then \[ \left|\ddy{\xb}{u}\times\ddy{\xb}{v}\right| = h_uh_v. \]
Proof. Recall that \(\displaystyle\ddy{\xb}{u}=h_u\eb_u\) and \(\displaystyle\ddy{\xb}{v}=h_v\eb_v\). Since \(h_u,h_v \geq 0\), we have that \[ \left|\ddy{\xb}{u}\times\ddy{\xb}{v}\right| = h_uh_v|\eb_u\times\eb_v| = h_uh_v|\pm\eb_w| = h_uh_v. \]
The surface area of \(S\) is computed by taking \(f(\xb)\equiv 1\), giving \(\displaystyle|S|=\int_S\,dS\).
Notice that the value of a scalar surface integral does not depend on the sign of \(\hat{\nb}\), so is independent of the choice of orientation of a surface (or indeed whether it is orientable at all).
Proposition 2.2 The surface integral of a scalar field is independent of the choice of parametrisation.
Proof. Suppose \(\xb(u,v)\) and \(\xb(\mu,\nu)\) are two regular parametrisations of the same surface. By the chain rule, \[ \ddy{\xb}{u} = \ddy{\xb}{\mu}\ddy{\mu}{u} + \ddy{\xb}{\nu}\ddy{\nu}{u}, \qquad \ddy{\xb}{v} = \ddy{\xb}{\mu}\ddy{\mu}{v} + \ddy{\xb}{\nu}\ddy{\nu}{v}. \] So \[\begin{align*} \ddy{\xb}{u}\times\ddy{\xb}{v} &= \left(\ddy{\xb}{\mu}\ddy{\mu}{u} + \ddy{\xb}{\nu}\ddy{\nu}{u}\right)\times\left(\ddy{\xb}{\mu}\ddy{\mu}{v} + \ddy{\xb}{\nu}\ddy{\nu}{v} \right)\\ &= \left(\ddy{\xb}{\mu}\ddy{\mu}{u} \times \ddy{\xb}{\nu}\ddy{\nu}{v}\right) + \left(\ddy{\xb}{\nu}\ddy{\nu}{u} \times \ddy{\xb}{\mu}\ddy{\mu}{v}\right)\\ &= \left(\ddy{\mu}{u}\ddy{\nu}{v} - \ddy{\mu}{v}\ddy{\nu}{u}\right)\ddy{\xb}{\mu}\times\ddy{\xb}{\nu}. \end{align*}\] This means \[ \left|\ddy{\xb}{u}\times\ddy{\xb}{v}\right| = |J|\left|\ddy{\xb}{\mu}\times\ddy{\xb}{\nu}\right|, \] where \(\displaystyle J= \ddy{\mu}{u}\ddy{\nu}{v} - \ddy{\mu}{v}\ddy{\nu}{u}\) is the Jacobian from Calculus I for the change of variables from \((\mu, \nu)\) to \((u,v)\). Since \(|J|\,du\,dv=d\mu\,d\nu\), we obtain \[ \int_Uf(\xb(u,v))\left|\ddy{\xb}{u}\times\ddy{\xb}{v}\right|\,du\,dv = \int_{U'}f(\xb(\mu,\nu))\left|\ddy{\xb}{\mu}\times\ddy{\xb}{\nu}\right|\,d\mu\,d\nu, \] where \(U'\) is the corresponding region of the \((\mu,\nu)\) plane.
If \(S\subset\Real^2\) is a flat surface in (say) the \((x,y)\)-plane, then \(\hat{\nb}\) only has a \(z\)-component, so \[ \left|\ddy{\xb}{u}\times\ddy{\xb}{v}\right| = \left|\ddy{x}{u}\ddy{y}{v} - \ddy{x}{v}\ddy{y}{u}\right| = |J|, \] which is just the Jacobian for a change of coordinates from \((x,y)\) to \((u,v)\) in the plane.
2.3 Surface integrals of vector fields
If \(\fb:\Real^3\to\Real^3\) is a vector field and \(S\) is an oriented surface with unit normal \(\hat{\nb}\), we define the surface integral (or flux) of \(\fb\) through \(S\) as \[ \int_S\fb\cdot\hat{\nb}\,dS. \]
Sometimes you will see the shorthand notation \(\dS = \hat{\nb}\,dS\) for the “vector area element”.
A non-zero flux requires a component of \(\fb\) pointing “through” the surface.
For given \(|\fb|\), the flux is largest if \(\fb\) is exactly perpendicular to the surface, but the flux also depends on the magnitude \(|\fb|\).
Again, we evaluate the integral using a specific parametrisation \(\xb(u,v)\), so \[\begin{align*} \int_S\fb\cdot\hat{\nb}\,dS &= \int_U \fb(\xb(u,v))\cdot\hat{\nb}\left|\ddy{\xb}{u}\times\ddy{\xb}{v}\right|\,du\,dv\\ &= \int_U \fb(\xb(u,v))\cdot\left(\ddy{\xb}{u}\times\ddy{\xb}{v}\right)\,du\,dv, \end{align*}\] where \(U\) is the preimage of \(S\) in the \((u,v)\) plane.
Unlike a scalar surface integral, a flux integral depends on the sign of \(\hat{\nb}\), i.e. on the orientation of the surface. You need to check that your chosen parametrisation \(\xb(u,v)\) gives the correct sign of \(\hat{\nb}=\displaystyle\ddy{\xb}{u}\times\ddy{\xb}{v}\). If not, interchange \(u\) and \(v\).
If the surface \(S\) is closed, this is often emphasised by writing \[ \int_S\fb\cdot\hat{\nb}\,dS = \oint_S\fb\cdot\hat{\nb}\,dS. \]
Unlike the circulation around a closed curve, there is no special name for the flux through a closed surface.
We will also meet vector-valued surface integrals of the form \[ \int_S\fb\,dS \] for some vector field \(\fb:\Real^3\to\Real^3\).
2.4 Volume integrals
The volume integral of a scalar field \(f:\Real^3\to\Real\) over a 3-d subregion \(V\subset\Real^3\) is \[ \int_Vf\,dV = \int_Uf(\xb(u,v,w))\left|\ddy{\xb}{u}\cdot\left(\ddy{\xb}{v}\times\ddy{\xb}{w}\right)\right|\,du\,dv\,dw, \] where \(U\) is the preimage of \(V\) in \((u,v,w)\) space, for some well-defined coordinates \((u,v,w)\).
The factor \(\displaystyle \left|\ddy{\xb}{u}\cdot\left(\ddy{\xb}{v}\times\ddy{\xb}{w}\right)\right|\,du\,dv\,dw\) is the volume of a parallelepiped in “real space” corresponding to an infinitesimal cuboid \((du,dv,dw)\) in \((u,v,w)\) space.
Proposition 2.3 If \((u,v,w)\) are orthogonal curvilinear coordinates, then \[ \left|\ddy{\xb}{u}\cdot\left(\ddy{\xb}{v}\times\ddy{\xb}{w}\right)\right| = h_uh_vh_w. \]
Proof. Recall that \(\displaystyle\ddy{\xb}{u}=h_u\eb_u\), and similarly for the other two tangent vectors. Since \(h_u,h_v,h_w \geq 0\), \[ \left|\ddy{\xb}{u}\cdot\left(\ddy{\xb}{v}\times\ddy{\xb}{w}\right)\right| = h_uh_vh_w |\eb_u\cdot(\eb_v\times\eb_w)| = h_uh_vh_w|\eb_u\cdot(\pm\eb_u)| = h_uh_vh_w. \]
The factor \(\displaystyle \left|\ddy{\xb}{u}\cdot\left(\ddy{\xb}{v}\times\ddy{\xb}{w}\right)\right|\) is often called the Jacobian (or Jacobian determinant). To see why, note that the scalar triple product may be written \[ \ddy{\xb}{u}\cdot\left(\ddy{\xb}{v}\times\ddy{\xb}{w}\right) = \begin{vmatrix} \displaystyle\ddy{x}{u} & \displaystyle\ddy{x}{v} & \displaystyle\ddy{x}{w}\\ \displaystyle\ddy{y}{u} & \displaystyle\ddy{y}{v} & \displaystyle\ddy{y}{w}\\ \displaystyle\ddy{z}{u} & \displaystyle\ddy{z}{v} & \displaystyle\ddy{z}{w} \end{vmatrix}. \] Compare this to the 2-d case of a flat surface in the \((x,y)\) plane, where we had \[ \left|\ddy{\xb}{u}\times\ddy{\xb}{v}\right| = \begin{vmatrix} \displaystyle\ddy{x}{u} & \displaystyle\ddy{x}{v}\\ \displaystyle\ddy{y}{u} & \displaystyle\ddy{y}{v} \end{vmatrix}. \]
The volume of \(V\) is computed by taking \(f(\xb)\equiv 1\), giving \(\displaystyle|V|=\int_V\,dV\).
In more complicated cases, the main difficulty is finding the correct integration limits. It is essential to draw a diagram!
Proposition 2.4 The volume integral of a scalar field is independent of the choice of coordinates.
Proof. This is analogous to Proposition 2.2, using the fact that a 3-d change of variables obeys \[ dx\,dy\,dz = \left|\ddy{\xb}{u}\cdot\left(\ddy{\xb}{v}\times\ddy{\xb}{w}\right)\right|\,du\,dv\,dw. \]
Occasionally you will come across a vector-valued volume integral, \(\displaystyle\int_V\fb\,dV\).
An example is the centre of mass of a solid body, \[ \overline{\xb} = \frac{\displaystyle\int_V\rho(\xb)\xb\,dV}{\displaystyle\int_V\rho(\xb)\,dV} \] where \(\rho(\xb)\) is the distribution of mass density within the body.