Surface integral and parametric surfaces

Intuition [Beginner]

A parametric surface is a curved sheet in 3D space described by a function $\mathbf{r}(u,v)$ that takes two parameters and outputs a point in ${\mathbb{R}}^3$. Think of it as a map from a flat region (the $uv$-plane) onto a curved surface.

The surface area of a parametric surface comes from the cross product of the two tangent vectors ${\mathbf{r}}_u$ and ${\mathbf{r}}_v$. The cross product gives a vector perpendicular to the surface whose length is the area of the tiny parallelogram spanned by the two tangent vectors.

The surface integral adds up a function $f$ over the surface, weighted by area. It generalises the line integral from curves to surfaces. If $f$ is the density of a thin shell, the surface integral gives the total mass.

Visual [Beginner]

A parametric surface $\mathbf{r}(u,v)$ shown as a curved grid in 3D. The grid lines come from fixing $u$ and varying $v$ (and vice versa). At one grid point, two tangent arrows ${\mathbf{r}}_u$ and ${\mathbf{r}}_v$ form a tiny parallelogram whose area is $dS$.

The surface integral: sum a function over tiny area patches of a curved surface.

Worked example [Beginner]

The sphere of radius $R$ can be parametrised by:

$$\mathbf{r}(\theta ,\varphi )=(R\sin \varphi \cos \theta ,R\sin \varphi \sin \theta ,R\cos \varphi )$$

for $0\le \theta \le 2\pi$ and $0\le \varphi \le \pi$.

The tangent vectors are ${\mathbf{r}}_{\theta }$ and ${\mathbf{r}}_{\varphi }$. Their cross product has magnitude $R^2\sin \varphi$.

The surface area is computed by adding up $R^2\sin \varphi$ over all $\theta$ from $0$ to $2\pi$ and $\varphi$ from $0$ to $\pi$. The result is $2\pi R^2\times 2=4\pi R^2$.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Definition (Parametric surface). A parametric surface in ${\mathbb{R}}^3$ is a smooth map $\mathbf{r}:D\to {\mathbb{R}}^3$ from a domain $D\subset {\mathbb{R}}^2$ such that the partial derivatives ${\mathbf{r}}_u=\partial \mathbf{r}/\partial u$ and ${\mathbf{r}}_v=\partial \mathbf{r}/\partial v$ are linearly independent at each point (the surface is regular).

Definition (Surface area element). The surface area element is:

$$dS=|{\mathbf{r}}_u\times {\mathbf{r}}_v|dudv.$$

Definition (Surface integral). The surface integral of a scalar function $f$ over the surface $S=\mathbf{r}(D)$ is:

$${\iint }_{S}fdS={\iint }_{D}f(\mathbf{r}(u,v))|{\mathbf{r}}_u\times {\mathbf{r}}_v|dudv.$$

For a vector field $\mathbf{F}$, the flux integral through $S$ with unit normal $\hat{\mathbf{n}}$ is:

$${\iint }_S\mathbf{F}\cdot \hat{\mathbf{n}}dS={\iint }_D\mathbf{F}(\mathbf{r}(u,v))\cdot ({\mathbf{r}}_u\times {\mathbf{r}}_v)dudv.$$

Key theorem with proof [Intermediate+]

Theorem (Surface area formula). If $\mathbf{r}:D\to {\mathbb{R}}^3$ is a regular parametric surface, then the surface area is:

$$\text{Area}(S)={\iint }_D|{\mathbf{r}}_u\times {\mathbf{r}}_v|dudv.$$

Proof. Partition $D$ into small rectangles $\Delta u\times \Delta v$. Each rectangle maps to a tiny patch on the surface. The image of the rectangle at $(u_0,v_0)$ is approximately the parallelogram spanned by ${\mathbf{r}}_u(u_0,v_0)\Delta u$ and ${\mathbf{r}}_v(u_0,v_0)\Delta v$.

The area of this parallelogram is $|{\mathbf{r}}_u\times {\mathbf{r}}_v|\Delta u\Delta v$. Summing over all patches and taking the limit gives the integral. The linear independence of ${\mathbf{r}}_u$ and ${\mathbf{r}}_v$ ensures the cross product is nonzero, so the area element is well-defined. $\square$

Bridge. This formula generalises the arc-length integral from curves to surfaces; the foundational reason it works is that the cross product encodes the area distortion of the parametrisation. This pattern appears again in the divergence theorem where the surface integral of a vector field is related to the volume integral of its divergence. The bridge is that the surface area element $dS$ is the 2-dimensional analogue of the arc-length element $ds$ — both measure how the parametrisation stretches area or length.

Exercises [Intermediate+]

Advanced results [Master]

The divergence theorem (Gauss). Let $V$ be a solid region in ${\mathbb{R}}^3$ with boundary surface $S=\partial V$ oriented outward, and let $\mathbf{F}$ be a smooth vector field on $V$. Then:

$${\iint }_S\mathbf{F}\cdot \hat{\mathbf{n}}dS={\iiint }_V\nabla \cdot \mathbf{F}dV.$$

This connects the flux through the boundary to the divergence inside the volume. It is the 3-dimensional case of Stokes' theorem and one of the four Maxwell equations in integral form.

The first fundamental form. For a parametric surface $\mathbf{r}(u,v)$, the first fundamental form is the symmetric matrix:

$$I=\left(\begin{array}{cc}E& F\\ F& G\end{array}\right)=\left(\begin{array}{cc}{\mathbf{r}}_u\cdot {\mathbf{r}}_u& {\mathbf{r}}_u\cdot {\mathbf{r}}_v\\ {\mathbf{r}}_u\cdot {\mathbf{r}}_v& {\mathbf{r}}_v\cdot {\mathbf{r}}_v\end{array}\right).$$

The surface area element is $dS=\sqrt{EG-F^2}dudv=|{\mathbf{r}}_u\times {\mathbf{r}}_v|dudv$, since $|{\mathbf{r}}_u\times {\mathbf{r}}_v{|}^2=EG-F^2$.

Area formula (geometric measure theory). The surface area is independent of the parametrisation: if ${\mathbf{r}}_1:D_1\to S$ and ${\mathbf{r}}_2:D_2\to S$ are two parametrisations of the same surface, they give the same area. This follows from the change-of-variables theorem and the Jacobian of the reparametrisation.

Synthesis. Surface integrals are the 2-dimensional generalisation of line integrals; the central insight is that the cross product provides the area distortion factor, just as the tangent vector provides the length distortion factor for curves. This pattern appears again in differential forms where ${\iint }_S\omega$ is defined on any $k$-dimensional surface for a $k$-form $\omega$. The bridge is that surface integrals build toward the general Stokes theorem ${\int }_{\partial M}\omega ={\int }_{M}d\omega$ which unifies the divergence theorem, Green's theorem, and the classical Stokes theorem into one result.

Full proof set [Master]

Proposition (Reparametrisation invariance). If $\mathbf{r}:D\to S$ and $\tilde{\mathbf{r}}:\tilde{D}\to S$ are two parametrisations of the same oriented surface $S$, then:

$${\iint }_{D}f(\mathbf{r}(u,v))|{\mathbf{r}}_u\times {\mathbf{r}}_v|dudv={\iint }_{\tilde{D}}f(\tilde{\mathbf{r}}(\tilde{u},\tilde{v}))|{\tilde{\mathbf{r}}}_{\tilde{u}}\times {\tilde{\mathbf{r}}}_{\tilde{v}}|d\tilde{u}d\tilde{v}.$$

Proof sketch. The reparametrisation $\mathbf{r}=\tilde{\mathbf{r}}\circ \Phi$ where $\Phi :D\to \tilde{D}$ is a diffeomorphism. By the chain rule, ${\mathbf{r}}_u\times {\mathbf{r}}_v=(\det D\Phi )({\tilde{\mathbf{r}}}_{\tilde{u}}\times {\tilde{\mathbf{r}}}_{\tilde{v}})\circ \Phi$. Then the change-of-variables theorem gives equality, with $|\det D\Phi |$ cancelling between the area element and the Jacobian of the substitution. $\square$

Connections [Master]

Line integrals generalise integration from intervals to curves; surface integrals extend this to 2-dimensional surfaces, and both are unified by integration on manifolds.

The divergence theorem relates the surface integral of a vector field to the volume integral of its divergence — it is the fundamental theorem of calculus for flux.

Stokes' theorem (the general form on manifolds) unifies the divergence theorem, Green's theorem, and the classical Stokes theorem into the single identity ${\int }_{\partial M}\omega ={\int }_{M}d\omega$.

Bibliography [Master]

@book{stewart-calculus,
  author = {Stewart, James},
  title = {Calculus: Early Transcendentals},
  edition = {9},
  publisher = {Cengage},
  year = {2020}
}

@book{apostol-calculus-v2,
  author = {Apostol, Tom M.},
  title = {Calculus},
  volume = {2},
  edition = {2},
  publisher = {Wiley},
  year = {1969}
}

@book{spivak-calculus-manifolds,
  author = {Spivak, Michael},
  title = {Calculus on Manifolds},
  publisher = {Benjamin},
  year = {1965}
}

Historical & philosophical context [Master]

The surface integral was developed in the early 19th century by Gauss ^{[Gauss 1813]}, who needed to compute gravitational flux through closed surfaces. The divergence theorem (also called Gauss's theorem or the Ostrogradsky-Gauss theorem) was proved independently by Gauss in 1813 and Ostrogradsky in 1826.

The general theory of integration on manifolds was completed by Elie Cartan in the 1890s using differential forms. The surface integral ${\iint }_{S}fdS$ is the special case of integrating a 2-form over a 2-dimensional surface. The modern treatment via parametric surfaces follows the exposition in Spivak's Calculus on Manifolds (1965), which made the theory accessible to undergraduates.