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Surface Integrals and Flux Worked Examples cheat sheet - grade college

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Calculus Grade college

Surface Integrals and Flux Worked Examples Cheat Sheet

A printable reference covering surface integrals, flux, parameterized surfaces, normal vectors, orientation, and the divergence theorem for college calculus.

TopicsIntegralsVectorsFunctionsDerivativesCalculusStudy Guide
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Surface integrals extend ordinary integration to curved surfaces in three-dimensional space. This cheat sheet helps students compute area integrals and flux integrals using parameterizations, normal vectors, and symmetry. It is especially useful for multivariable calculus problems involving fluids, electric fields, heat flow, and vector fields through surfaces.

Worked-example patterns make it easier to choose the right setup before doing algebra.

The core idea is that a small patch of surface has vector area r⃗u×r⃗v du dv\vec{r}_u \times \vec{r}_v\,du\,dv, where r⃗(u,v)\vec{r}(u,v) parameterizes the surface. A scalar surface integral uses dS=∥r⃗u×r⃗v∥du dvdS = \left\lVert \vec{r}_u \times \vec{r}_v \right\rVert du\,dv, while flux uses F⃗⋅n⃗ dS\vec{F} \cdot \vec{n}\,dS. For closed surfaces, the divergence theorem often changes a difficult flux integral into a triple integral, ∬SF⃗⋅n⃗ dS=∭E∇⋅F⃗ dV\iint_S \vec{F} \cdot \vec{n}\,dS = \iiint_E \nabla \cdot \vec{F}\,dV.

Orientation matters because reversing the normal vector changes the sign of flux.

Key Facts

  • For a parameterized surface r⃗(u,v)\vec{r}(u,v), the scalar surface element is dS=∥r⃗u×r⃗v∥du dvdS = \left\lVert \vec{r}_u \times \vec{r}_v \right\rVert du\,dv.
  • A scalar surface integral is computed by ∬Sf(x,y,z) dS=∬Df(r⃗(u,v))∥r⃗u×r⃗v∥du dv\iint_S f(x,y,z)\,dS = \iint_D f(\vec{r}(u,v))\left\lVert \vec{r}_u \times \vec{r}_v \right\rVert du\,dv.
  • A flux integral through an oriented surface is ∬SF⃗⋅n⃗ dS=∬DF⃗(r⃗(u,v))⋅(r⃗u×r⃗v) du dv\iint_S \vec{F} \cdot \vec{n}\,dS = \iint_D \vec{F}(\vec{r}(u,v)) \cdot (\vec{r}_u \times \vec{r}_v)\,du\,dv when r⃗u×r⃗v\vec{r}_u \times \vec{r}_v gives the chosen orientation.
  • For a graph z=g(x,y)z = g(x,y) with upward orientation, a normal vector area element is ⟨−gx,−gy,1⟩ dx dy\langle -g_x, -g_y, 1 \rangle\,dx\,dy.
  • For a graph z=g(x,y)z = g(x,y), the scalar surface element is dS=1+gx2+gy2 dx dydS = \sqrt{1 + g_x^2 + g_y^2}\,dx\,dy.
  • The divergence theorem states that ∬SF⃗⋅n⃗ dS=∭E∇⋅F⃗ dV\iint_S \vec{F} \cdot \vec{n}\,dS = \iiint_E \nabla \cdot \vec{F}\,dV for a closed, outward-oriented surface SS enclosing a solid region EE.
  • The divergence of F⃗=⟨P,Q,R⟩\vec{F} = \langle P,Q,R \rangle is ∇⋅F⃗=∂P∂x+∂Q∂y+∂R∂z\nabla \cdot \vec{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}.
  • Reversing the orientation of a surface changes the flux from ∬SF⃗⋅n⃗ dS\iint_S \vec{F} \cdot \vec{n}\,dS to −∬SF⃗⋅n⃗ dS-\iint_S \vec{F} \cdot \vec{n}\,dS.

Vocabulary

Surface integral
An integral that sums a scalar or vector quantity over a surface in three-dimensional space.
Flux
The signed amount of a vector field passing through an oriented surface, computed with ∬SF⃗⋅n⃗ dS\iint_S \vec{F} \cdot \vec{n}\,dS.
Parameterization
A vector function r⃗(u,v)\vec{r}(u,v) that describes every point on a surface using two parameters.
Normal vector
A vector perpendicular to a surface, often found from r⃗u×r⃗v\vec{r}_u \times \vec{r}_v.
Orientation
The chosen direction of the normal vector on a surface, such as upward, downward, inward, or outward.
Divergence theorem
A theorem that converts outward flux through a closed surface into a triple integral of divergence over the enclosed solid.

Common Mistakes to Avoid

  • Using ∥r⃗u×r⃗v∥\left\lVert \vec{r}_u \times \vec{r}_v \right\rVert for flux is wrong because flux needs the signed vector area element, not just surface area.
  • Ignoring orientation gives the wrong sign because r⃗u×r⃗v\vec{r}_u \times \vec{r}_v may point opposite the required normal direction.
  • Forgetting to substitute the parameterization into F⃗\vec{F} or ff is wrong because the integrand must be written in terms of the variables of integration.
  • Applying the divergence theorem to an open surface is wrong unless the surface is first closed by adding the missing boundary pieces and accounting for their flux.
  • Using dAdA instead of dSdS in a scalar surface integral is wrong for slanted or curved surfaces because dSdS includes the stretching factor 1+gx2+gy2\sqrt{1 + g_x^2 + g_y^2}.

Practice Questions

  1. 1 Compute ∬Sz dS\iint_S z\,dS for the plane z=2x+yz = 2x + y above the rectangle 0≤x≤10 \le x \le 1, 0≤y≤20 \le y \le 2.
  2. 2 Find the upward flux of F⃗=⟨x,y,z⟩\vec{F} = \langle x,y,z \rangle through the surface z=4−x2−y2z = 4 - x^2 - y^2 over the disk x2+y2≤4x^2 + y^2 \le 4.
  3. 3 Use the divergence theorem to find the outward flux of F⃗=⟨x2,y2,z2⟩\vec{F} = \langle x^2,y^2,z^2 \rangle through the cube 0≤x≤10 \le x \le 1, 0≤y≤10 \le y \le 1, 0≤z≤10 \le z \le 1.
  4. 4 Explain why reversing the orientation of a surface changes the sign of a flux integral but does not change a scalar surface integral.