6.3 Surface Area and Parametric Surfaces

Parameterization helps us describe 3D surfaces using 2D parameters. We can map points on a plane to points on a surface, making it easier to analyze complex shapes. This technique is crucial for understanding the geometry of curved surfaces in space.

Surface area calculations build on parameterization. By using vector-valued functions and partial derivatives, we can find the area of curved surfaces through integration. This connects our understanding of surfaces to calculus concepts we've learned before.

Parameterization and Surface Area

Parameterization of 3D surfaces

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• Vector-valued functions for surfaces map 2D parameters to 3D points $\mathbf{r}(u,v) = \langle x(u,v), y(u,v), z(u,v) \rangle$ defined over a domain in uv-plane
• Common surface types with parameterizations:
  • Planes: $\mathbf{r}(u,v) = \mathbf{p} + u\mathbf{v} + v\mathbf{w}$ where $\mathbf{p}$ is a point on the plane and $\mathbf{v}$, $\mathbf{w}$ are vectors in the plane
  • Spheres: $\mathbf{r}(\theta,\phi) = \langle a\sin\phi\cos\theta, a\sin\phi\sin\theta, a\cos\phi \rangle$ with $a$ as radius, $\theta$ as azimuthal angle, $\phi$ as polar angle
  • Cylinders: $\mathbf{r}(\theta,z) = \langle a\cos\theta, a\sin\theta, z \rangle$ where $a$ is radius and $z$ is height
  • Cones: $\mathbf{r}(\theta,z) = \langle z\tan\alpha\cos\theta, z\tan\alpha\sin\theta, z \rangle$ with $\alpha$ as half-angle of cone
• Identifying parameters involves recognizing surface symmetry and natural coordinates (radial, angular)
• Transforming between Cartesian and parametric forms requires understanding geometric relationships and solving equations

Surface area of parametric surfaces

• Surface integral calculates area by summing infinitesimal patches over entire surface
• Fundamental formula $SA = \iint_D |\mathbf{r}_u \times \mathbf{r}_v| \, du \, dv$ uses partial derivatives and cross product
• Surface area calculation process:
  1. Determine parameterization $\mathbf{r}(u,v)$ based on surface geometry
  2. Compute partial derivatives $\mathbf{r}_u$ and $\mathbf{r}_v$ using component-wise differentiation
  3. Calculate cross product $\mathbf{r}_u \times \mathbf{r}_v$ representing surface normal vector
  4. Find magnitude $|\mathbf{r}_u \times \mathbf{r}_v|$ using dot product or component formula
  5. Set up double integral over parameter domain D and evaluate using appropriate techniques (substitution, numerical methods)

Surface area formula derivation

• Jacobian matrix $J = \begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} \end{bmatrix}$ encodes information about surface deformation
• Relationship $\mathbf{r}_u \times \mathbf{r}_v = \det(J)$ connects cross product to Jacobian determinant
• Derivation steps:
  1. Approximate surface with small parallelograms formed by $\mathbf{r}_u \Delta u$ and $\mathbf{r}_v \Delta v$
  2. Area of parallelogram given by magnitude of cross product: $|\mathbf{r}_u \times \mathbf{r}_v| \Delta u \Delta v$
  3. Take limit as $\Delta u, \Delta v \to 0$ to get infinitesimal area element
  4. Express result using Jacobian: $dA = \sqrt{\det(J^T J)} \, du \, dv$
• Final surface area formula $SA = \iint_D \sqrt{\det(J^T J)} \, du \, dv$ generalizes cross product form

Surface area in coordinate systems

• Cartesian coordinates for surfaces $z = f(x,y)$:
  • Parameterization $\mathbf{r}(x,y) = \langle x, y, f(x,y) \rangle$ uses x, y as parameters
  • Surface area formula $SA = \iint_D \sqrt{1 + (\frac{\partial f}{\partial x})^2 + (\frac{\partial f}{\partial y})^2} \, dx \, dy$ derived from general form
• Spherical coordinates:
  • Conversion $x = \rho\sin\phi\cos\theta, y = \rho\sin\phi\sin\theta, z = \rho\cos\phi$ with $\rho$ as radius, $\theta$ as azimuthal angle, $\phi$ as polar angle
  • Surface area element $dS = \rho^2\sin\phi \, d\theta \, d\phi$ useful for surfaces with spherical symmetry
• Cylindrical coordinates:
  • Conversion $x = r\cos\theta, y = r\sin\theta, z = z$ with $r$ as radial distance, $\theta$ as angular coordinate
  • Surface area element $dS = r \, d\theta \, dz$ simplifies calculations for cylindrical surfaces
• Choosing coordinate system based on surface symmetry reduces complexity of integration
• Changing variables in surface integrals involves Jacobian of transformation between coordinate systems