5️⃣Multivariable Calculus Unit 7 – Surface Integrals & Stokes' Theorem

Key Concepts

• Surface integrals extend the concept of line integrals to surfaces in three-dimensional space
• Surfaces can be parametrized using two variables, typically denoted as $u$ and $v$
• The surface integral of a scalar function $f(x, y, z)$ over a surface $S$ is denoted as $\iint_S f(x, y, z) dS$
• The surface integral of a vector field $\mathbf{F}(x, y, z)$ over a surface $S$ is denoted as $\iint_S \mathbf{F} \cdot d\mathbf{S}$ or $\iint_S \mathbf{F} \times d\mathbf{S}$
  • $\mathbf{F} \cdot d\mathbf{S}$ represents the flux of $\mathbf{F}$ through the surface $S$
  • $\mathbf{F} \times d\mathbf{S}$ represents the circulation of $\mathbf{F}$ along the surface $S$
• Stokes' theorem relates the surface integral of the curl of a vector field over a surface to the line integral of the vector field along the boundary of the surface
• Stokes' theorem is a generalization of Green's theorem to three-dimensional space

Geometric Interpretation

• Surface integrals can be visualized as the sum of the values of a function over a surface, similar to how line integrals sum values along a curve
• The surface integral of a scalar function $f(x, y, z)$ can be interpreted as the volume under the surface $z = f(x, y)$ above the $xy$-plane
• The flux of a vector field $\mathbf{F}$ through a surface $S$ represents the amount of "flow" of $\mathbf{F}$ through $S$
  • Positive flux indicates flow in the direction of the surface normal, while negative flux indicates flow opposite to the surface normal
• The circulation of a vector field $\mathbf{F}$ along a surface $S$ measures the tendency of $\mathbf{F}$ to rotate around the surface
• Stokes' theorem relates the circulation of a vector field along the boundary of a surface to the flux of the curl of the vector field through the surface
  • This can be visualized as the net flow around the boundary being equal to the total "swirling" of the vector field within the surface

Types of Surface Integrals

• Scalar surface integrals: $\iint_S f(x, y, z) dS$
  • Integrate a scalar function $f(x, y, z)$ over a surface $S$
  • Can be used to find the area of a surface by setting $f(x, y, z) = 1$
• Vector surface integrals of the first kind (flux): $\iint_S \mathbf{F} \cdot d\mathbf{S}$
  • Integrate the dot product of a vector field $\mathbf{F}$ with the surface normal $d\mathbf{S}$
  • Measures the flux of $\mathbf{F}$ through the surface $S$
• Vector surface integrals of the second kind (circulation): $\iint_S \mathbf{F} \times d\mathbf{S}$
  • Integrate the cross product of a vector field $\mathbf{F}$ with the surface normal $d\mathbf{S}$
  • Measures the circulation of $\mathbf{F}$ along the surface $S$
• Surface integrals can be evaluated over parametric surfaces, where the surface is defined by a vector-valued function $\mathbf{r}(u, v)$

Calculating Surface Integrals

• To evaluate a surface integral, parametrize the surface using a vector-valued function $\mathbf{r}(u, v) = (x(u, v), y(u, v), z(u, v))$
• Compute the partial derivatives $\mathbf{r}_u$ and $\mathbf{r}_v$
• Calculate the cross product $\mathbf{r}_u \times \mathbf{r}_v$ to find the surface normal vector $\mathbf{n}$
  • The magnitude of $\mathbf{r}_u \times \mathbf{r}_v$ gives the surface area element $dS$
• Substitute the parametrization into the integrand and evaluate the integral using the surface area element $dS$
  • For scalar surface integrals: $\iint_S f(x, y, z) dS = \iint_D f(\mathbf{r}(u, v)) \|\mathbf{r}_u \times \mathbf{r}_v\| du dv$
  • For vector surface integrals of the first kind: $\iint_S \mathbf{F} \cdot d\mathbf{S} = \iint_D \mathbf{F}(\mathbf{r}(u, v)) \cdot (\mathbf{r}_u \times \mathbf{r}_v) du dv$
  • For vector surface integrals of the second kind: $\iint_S \mathbf{F} \times d\mathbf{S} = \iint_D \mathbf{F}(\mathbf{r}(u, v)) \times (\mathbf{r}_u \times \mathbf{r}_v) du dv$
• Determine the limits of integration based on the parametrization and evaluate the resulting double integral

Stokes' Theorem

• Stokes' theorem relates the surface integral of the curl of a vector field over a surface to the line integral of the vector field along the boundary of the surface
• Mathematically, Stokes' theorem is stated as: $\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \oint_C \mathbf{F} \cdot d\mathbf{r}$
  • $S$ is a smooth, orientable surface bounded by a simple, closed curve $C$
  • $\mathbf{F}$ is a vector field defined on $S$ and $C$
  • $\nabla \times \mathbf{F}$ is the curl of $\mathbf{F}$
  • $d\mathbf{S}$ is the surface area element, and $d\mathbf{r}$ is the line element along $C$
• Stokes' theorem allows for the conversion between surface integrals and line integrals, which can simplify calculations in certain situations
• To apply Stokes' theorem, ensure that the surface is smooth, orientable, and bounded by a simple, closed curve
  • Choose the orientation of the surface normal consistent with the right-hand rule relative to the boundary curve orientation

Applications

• Surface integrals have numerous applications in physics, engineering, and other fields
• Flux calculations: Surface integrals can be used to calculate the flux of a vector field through a surface
  • Examples include electric flux in electromagnetism and fluid flow through a surface in fluid dynamics
• Circulation calculations: Surface integrals can be used to calculate the circulation of a vector field along a surface
  • Examples include the circulation of a fluid along a surface in fluid dynamics and the circulation of an electromagnetic field in electromagnetism
• Surface area calculations: Setting the integrand to 1 in a scalar surface integral yields the surface area of the given surface
• Stokes' theorem can be used to simplify calculations by converting between surface integrals and line integrals
  • For example, calculating the work done by a force field along a closed path using a line integral instead of a surface integral

Common Pitfalls

• Incorrectly parametrizing the surface, leading to incorrect surface normal vectors and surface area elements
• Inconsistent orientation of the surface normal and boundary curve when applying Stokes' theorem
  • The orientation of the surface normal should follow the right-hand rule relative to the boundary curve orientation
• Forgetting to include the magnitude of the cross product $\|\mathbf{r}_u \times \mathbf{r}_v\|$ as the surface area element when evaluating surface integrals
• Misinterpreting the meaning of the flux and circulation integrals
  • Flux measures the flow through a surface, while circulation measures the tendency to rotate along a surface
• Attempting to apply Stokes' theorem to non-smooth, non-orientable, or unbounded surfaces
• Incorrectly calculating the curl of a vector field when applying Stokes' theorem
• Mixing up the order of integration or the limits of integration when evaluating surface integrals

Practice Problems

1. Evaluate the scalar surface integral $\iint_S (x^2 + y^2) dS$, where $S$ is the portion of the paraboloid $z = x^2 + y^2$ that lies above the square $0 \leq x \leq 1$, $0 \leq y \leq 1$ in the $xy$-plane.

2. Calculate the flux of the vector field $\mathbf{F}(x, y, z) = (x, y, z)$ through the surface $S$, which is the portion of the sphere $x^2 + y^2 + z^2 = 1$ that lies above the $xy$-plane.

3. Evaluate the circulation of the vector field $\mathbf{F}(x, y, z) = (y, -x, 0)$ along the surface of the cone $z = \sqrt{x^2 + y^2}$, bounded by the plane $z = 1$.

4. Use Stokes' theorem to evaluate $\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}$, where $\mathbf{F}(x, y, z) = (x^2, y^2, z^2)$ and $S$ is the portion of the plane $2x + y + z = 1$ that lies in the first octant, oriented upward.

5. Find the area of the surface $z = \sin(x) + \cos(y)$ over the region $D = \{(x, y) : 0 \leq x \leq \pi, 0 \leq y \leq \pi\}$.

6. Verify Stokes' theorem for the vector field $\mathbf{F}(x, y, z) = (y, -x, 0)$ and the surface $S$, which is the portion of the paraboloid $z = 1 - x^2 - y^2$ that lies above the $xy$-plane, oriented upward.

7. Calculate the flux of the vector field $\mathbf{F}(x, y, z) = (x, y, z)$ through the surface $S$, which is the portion of the cylinder $x^2 + y^2 = 1$ bounded by the planes $z = 0$ and $z = 2$, oriented outward.

8. Evaluate the surface integral $\iint_S (x + y + z) dS$, where $S$ is the surface of the tetrahedron with vertices at $(0, 0, 0)$, $(1, 0, 0)$, $(0, 1, 0)$, and $(0, 0, 1)$.