Calculus of Parametric Surfaces and Stokes' Theorem

Review of Green's Theorem and Vector Operators

• Green's Theorem Definition: As established previously, if a curve $C$ is the boundary of a domain $D$, the line integral along $C$ is equal to the double integral over $D$ of the difference of partial derivatives.
• Formula:
  $\oint_C P\,dx + Q\,dy = \iint_D (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y})\,dA$
• Orientation Requirements:
      * For the theorem to hold, the domain must stay on the left side as one walks along the boundary.
      * The outer boundary of the domain must be oriented counterclockwise.
      * Any inner boundaries (holes) must be oriented clockwise.
• Del Operator (Derivative Operator): Represented by $\nabla$, it consists of partial derivatives:
  $\nabla = \langle \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \rangle$
• Curl of a Vector Field: Defined as the cross product of the del operator and the vector field $\mathbf{F}$, denoted as $\text{curl}\,\mathbf{F}$ or $\nabla \times \mathbf{F}$.
• Divergence of a Vector Field: Defined as the dot product of the del operator and the vector field $\mathbf{F}$, denoted as $\text{div}\,\mathbf{F}$ or $\nabla \cdot \mathbf{F}$.

Parametric Surfaces and Vector Representations (Section 16.6)

• Limitations of Graph-based Surfaces: Previously, surfaces were treated as graphs of functions where $z = g(x, y)$. Such surfaces represent a collection of points of the form $\langle x, y, g(x, y) \rangle$ lying above a domain $D$ in the $xy$-plane.
• The Vertical Line Test Problem: Not all surfaces can be expressed as $z = g(x, y)$. For example, a shape that doubles back on itself fails the vertical line test and cannot be written as a single function of $x$ and $y$.
• Conceptual Mapping: A two-dimensional object should be parametrizable using two variables (coordinates). We define coordinates $u$ and $v$ in a domain $D$ within the $uv$-plane.
• Parametric Representation: We map points from the $uv$-plane into three-dimensional space ($xyz$) using a vector-valued function:
  $\mathbf{r}(u, v) = \langle x(u, v), y(u, v), z(u, v) \rangle$
• Coordinate Grid Mapping: This map takes a two-dimensional grid from the coordinate space and deforms/places it into 3D space to form a surface.

Area of a Parametric Surface

• Differential Area Approximation: To find the surface area, the domain in the $uv$-plane is subdivided into small rectangles with sides $\Delta u$ and $\Delta v$. These rectangles are mapped to "approximate parallelograms" on the surface in 3D space.
• Tangent Vectors: The sides of these approximate parallelograms are represented by vectors determined by partial derivatives of $\mathbf{r}$.
  $\mathbf{r}_u = \langle \frac{\partial x}{\partial u}, \frac{\partial y}{\partial u}, \frac{\partial z}{\partial u} \rangle$
  $\mathbf{r}_v = \langle \frac{\partial x}{\partial v}, \frac{\partial y}{\partial v}, \frac{\partial z}{\partial v} \rangle$
• Geometric Scaling: The actual lengths of the vectors forming the parallelogram on the surface are approximately $\mathbf{r}_u \Delta u$ and $\mathbf{r}_v \Delta v$
• Area Calculation: The area of a parallelogram defined by two vectors is the magnitude of their cross product.
  $\Delta S \approx |(\mathbf{r}_u \Delta u) \times (\mathbf{r}_v \Delta v)| = |\mathbf{r}_u \times \mathbf{r}_v| \Delta u \Delta v$
• Surface Area Formula: By taking the limit as subdivisions become finer, we convert the sum into an integral over the domain $D$ in the $uv$-space.
  $A(S) = \iint_D |\mathbf{r}_u \times \mathbf{r}_v|\,dA$
  $\text{where } dA = du\,dv$

Surface Integrals of Scalar Functions (Section 16.7)

• Definition: A surface integral extends the concept of a line integral to a two-dimensional surface. It integrates a scalar function $f(x, y, z)$ over the surface $S$.
• Transformation to u,v Space: The function is evaluated at points on the surface using the parametrization $\mathbf{r}(u, v)$, and then scaled by the surface area element.
  $f(x, y, z) = f(\mathbf{r}(u, v))$
• Integral Formulation:
  $\iint_S f(x, y, z)\,dS = \iint_D f(\mathbf{r}(u, v)) |\mathbf{r}_u \times \mathbf{r}_v|\,dA$
• Physical Interpretation: This represents taking the value of the function at a point on the surface and multiplying it by the area of the corresponding infinitesimal rectangle on the surface.

Verification with Surfaces as Graphs

• Special Case: $z = g(x, y)$: To verify the general formula, we apply it to a surface defined as a graph.
• Parametrization of a Graph:
  $u = x$
  $v = y$
  $z = g(x, y)$
  $\mathbf{r}(x, y) = \langle x, y, g(x, y) \rangle$
• Partial Derivatives:
  $\mathbf{r}_x = \langle 1, 0, g_x \rangle$
  $\mathbf{r}_y = \langle 0, 1, g_y \rangle$
• Cross Product Calculation:
  $\mathbf{r}_x \times \mathbf{r}_y = \langle -g_x, -g_y, 1 \rangle$
• Norm of the Cross Product:
  $|\mathbf{r}_x \times \mathbf{r}_y| = \sqrt{(-g_x)^2 + (-g_y)^2 + 1^2} = \sqrt{1 + g_x^2 + g_y^2}$
• Sanity Check: This result matches the formula for surface area derived in previous chapters, confirming the validity of the general parametric approach.

Vector Fields and the Concept of Flux

• Physical Motivation: Flux originates from fluid dynamics (measuring the rate of fluid flow) and electromagnetism.
• Flux through a Rectangular Surface: Suppose a vector field $\mathbf{F}$ represents fluid velocity. If a surface is parallel to the flow, zero fluid passes through (flux = 0). If the surface is perpendicular to the flow, maximum fluid passes through.
• The Dot Product Relationship: We need a function of the velocity $\mathbf{F}$ and the surface normal $\mathbf{n}$ that is zero when perpendicular, maximum when parallel, and negative when flow is opposite the normal. The dot product $\mathbf{F} \cdot \mathbf{n}$ satisfies these requirements.
• Flux Definition: Roughly, the rate of fluid flow through a surface.

Evaluating Flux Integrals

• Orientation and Normal Vectors: Surfaces have two sides. An orientation is a consistent choice of unit normal vector $\mathbf{n}$ across the surface. While not all surfaces are orientable (e.g., Mobius strip), standard surfaces like spheres are.
• Integral Setup:
  $\iint_S \mathbf{F} \cdot \mathbf{n}\,dS$
• Computing with Parametrization: The normal vector is given by the cross product of the tangent vectors $\mathbf{r}_u \times \mathbf{r}_v$. To make it a unit normal $\mathbf{n}$, we divide by its norm:
  $\mathbf{n} = \frac{\mathbf{r}_u \times \mathbf{r}_v}{|\mathbf{r}_u \times \mathbf{r}_v|}$
  $dS = |\mathbf{r}_u \times \mathbf{r}_v|\,dA$
• Resulting Formula: The norms cancel out, simplifying the calculation.
  $\iint_S \mathbf{F} \cdot d\mathbf{S} = \iint_D \mathbf{F}(\mathbf{r}(u, v)) \cdot (\mathbf{r}_u \times \mathbf{r}_v)\,dA$
• Notation Variants:
      * $\iint_S \mathbf{F} \cdot \mathbf{n}\,dS$
      * $\iint_S \mathbf{F} \cdot d\mathbf{S}$

Flux over a Graph of a Function

• Special Case Component Calculation: For $z = g(x, y)$, with $\mathbf{F} = \langle P, Q, R \rangle$.
• Vectors:
  $\text{Surface Normal (non-unit)} = \langle -g_x, -g_y, 1 \rangle$
• The Dot Product:
  $\mathbf{F} \cdot (\mathbf{r}_x \times \mathbf{r}_y) = \langle P, Q, R \rangle \cdot \langle -g_x, -g_y, 1 \rangle = -Pg_x - Qg_y + R$
• Full Formula:
  $\iint_S \mathbf{F} \cdot d\mathbf{S} = \iint_D (-P g_x - Q g_y + R)\,dA$

Stokes' Theorem (Section 16.8)

• Statement: Stokes' Theorem equates the line integral of a vector field along the boundary of a surface to the flux integral of the curl of that vector field through the surface.
• Formula:
  $\oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}$
• Orientation and the Right-Hand Rule: The orientation of the boundary curve must be consistent with the surface normal. If your thumb points in the direction of the normal, your fingers curl in the positive direction of the boundary curve.
• Historical and Experimental Context: This theorem was not immediately obvious but emerged through experimental observations in electromagnetism (e.g., Faraday's Law of Induction). Physicists noticed that different physical calculations yielded the same values repeatedly, leading to the formalization of the theorem.

Proof of Stokes' Theorem for Function Graphs

• Case Assumption: The surface is the graph of a function $z = g(x, y)$.
• Step 1: Compute the Right-Hand Side (Curl Flux):
      * Curl of $\mathbf{F}$:
  $\nabla \times \mathbf{F} = \langle R_y - Q_z, P_z - R_x, Q_x - P_y \rangle$
      * Apply the flux formula over a graph:
  $\iint_S \text{curl}\,\mathbf{F} \cdot d\mathbf{S} = \iint_D (-(R_y - Q_z)g_x - (P_z - R_x)g_y + (Q_x - P_y))\,dA$
• Step 2: Compute the Left-Hand Side (Line Integral):
      * Parameterize the boundary curve as $\langle x(t), y(t), z(t) \rangle$, where $z(t) = g(x(t), y(t))$.
      * By the chain rule:
  $z'(t) = g_x x'(t) + g_y y'(t)$
      * Substitute into the line integral $\int P\,dx + Q\,dy + R\,dz$:
  $\int P x' + Q y' + R(g_x x' + g_y y')\,dt$
  $\int (P + Rg_x)x'\,dt + (Q + Rg_y)y'\,dt$
      * This is a 2D line integral on the boundary of $D$ in the $xy$-plane. Apply Green's Theorem:
  $\iint_D (\frac{\partial}{\partial x}(Q + Rg_y) - \frac{\partial}{\partial y}(P + Rg_x))\,dA$
• Step 3: Expand the Derivatives (Chain Rule Application):
      * When differentiating functions like $Q(x, y, g(x, y))$ with respect to $x$, we must account for the implicit dependence of $z$ on $x$:
  $\frac{\partial Q}{\partial x} \rightarrow Q_x + Q_z g_x$
      * Expanding the full expression:
  $(Q_x + Q_z g_x + R_x g_y + R_z g_x g_y + R g_{xy}) - (P_y + P_z g_y + R_y g_x + R_z g_y g_x + R g_{yx})$
• Step 4: Cancel and Compare:
      * The terms $R_z g_x g_y$ and $R g_{xy}$ cancel out.
      * The resulting expression is:
  $Q_x + Q_z g_x + R_x g_y - P_y - P_z g_y - R_y g_x$
      * Distributing the terms in the RHS expression from Step 1:
  $-R_y g_x + Q_z g_x - P_z g_y + R_x g_y + Q_x - P_y$
• Conclusion: Both sides are identical, proving the theorem for this case.