Problem 13 - Thomas Calculus

Question

Let \(n\) be the outer unit normal of the elliptical shell $$4 x^{2}+9 y^{2}+36 z^{2}=36, \quad z \geq 0$$ and let $$\mathbf{F}=y \mathbf{i}+x^{2} \mathbf{j}+\left(x^{2}+y^{4}\right)^{3 / 2} \sin e^{\sqrt{x z}} \mathbf{k}$$ Find the value of $$\iint_{S}(\nabla \times \mathbf{F}) \cdot \mathbf{n} d \sigma$$ (Hint: One parametrization of the ellipse at the base of the shell is \(x=3 \cos t, y=2 \sin t, 0 \leq t \leq 2 \pi . )\)

Step-by-Step Solution

Answer

The integral \( \iint_{S}(\nabla \times \mathbf{F}) \cdot \mathbf{n} d \sigma = 0 \).

Step 1: Identify the Surface

The elliptical shell has the equation \(4x^2 + 9y^2 + 36z^2 = 36\) with \(z \geq 0\). This describes an ellipsoid centered at the origin, truncated at \(z = 0\). We want to find the integral over its upper half surface, denoted as \(S\).

Step 2: Simplify using Stokes' Theorem

By Stokes' Theorem, the surface integral \( \iint_{S}(abla \times \mathbf{F}) \cdot \mathbf{n} \, d\sigma \) over the closed surface \( S \) is equivalent to the line integral of \( \mathbf{F} \) over the boundary curve of the base. The boundary of the upper half is given by the ellipse \( x=3\cos t, y=2\sin t, z=0 \).

Step 3: Parametrize the Boundary

The parameterization for the elliptical boundary is given as \( \mathbf{r}(t) = 3\cos t \, \mathbf{i} + 2\sin t \, \mathbf{j} \) for \( 0 \leq t \leq 2\pi \).

Step 4: Evaluate \(\mathbf{F}(x, y, z)\) along the Boundary

Substitute \( x = 3\cos t \) and \( y = 2\sin t \) into the vector field \( \mathbf{F} \). At the boundary (since \( z = 0 \)), the expression reduces to:

Step 5: Integrate Over Boundary

Parametrize the line integral for \( \oint_{C} \mathbf{F} \cdot d\mathbf{r} \) using \( \mathbf{r}(t) \) and evaluate over \( t \) from \( 0 \) to \( 2\pi \).

Step 6: Apply Stoke's Theorem Result

We don't need the actual computation for \( \mathbf{F} \cdot d\mathbf{r} \) along the boundary, because \( abla \times \mathbf{F} = 0 \). Thus, the integral \( \iint_{S}(abla \times \mathbf{F}) \cdot \mathbf{n} d \sigma = 0 \) by Stoke's Theorem.

Key Concepts

Elliptical Shell

An elliptical shell is a 3D shape derived from an ellipsoid equation such that it is open on one side. In this exercise, we consider the shell defined by the equation \(4x^2 + 9y^2 + 36z^2 = 36\), focusing on its top half where \(z \geq 0\). This equation describes an ellipsoid centered at the origin with semi-axes length along the x, y, and z axes defined by the constants in the equations. Here, these lengths are 3, 2, and 1 respectively. Understanding the geometry helps comprehend how integrals are evaluated over this surface. When the problem talks about the 'outer unit normal', it refers to a vector that is perpendicular to the surface at a given point, pointing outward.

Vector Fields

Vector fields are mathematical constructions used to assign a vector to each point in a space. They can be visualized like a collection of arrows, each pointing in a specific direction and having magnitude. In our context, the vector field \(\mathbf{F} = y \mathbf{i} + x^2 \mathbf{j} + (x^2 + y^4)^{3/2} \sin e^{\sqrt{x z}} \mathbf{k}\) is assigned to the space in and around the elliptical shell. Breaking it down:

• \(y \mathbf{i}\) implies the vector has a component influenced by the y-coordinate along the x-direction, represented by \(\mathbf{i}\).
• \(x^2 \mathbf{j}\) indicates a magnitude derived from the x-coordinate along the y-direction, represented by \(\mathbf{j}\).
• Similarly, \((x^2 + y^4)^{3/2} \sin e^{\sqrt{x z}} \mathbf{k}\) complicates things further by being a function of both x and y, depending on additional trigonometric and exponential functions along the z-direction represented by \(\mathbf{k}\).

Understanding each component's influence is vital for evaluating integrals in vector field contexts.

Surface Integrals

The concept of a surface integral is an extension of line integrals to two dimensions, allowing for integration over a surface in a 3D space. This is done by summing quantities over all points on a surface to get a total value. In this exercise, we compute \(\iint_{S}(abla \times \mathbf{F}) \cdot \mathbf{n} \, d\sigma\), where we project vector field rotations (\(abla \times \mathbf{F}\)) onto the surface \(S\), against the unit normal vector \(\mathbf{n}\).
Surface integrals become especially relevant in physical applications and help us comprehend how vector fields "flow" across surfaces. Stokes' Theorem links these integrals to line integrals around the boundary of the surface, simplifying calculations and solving problems like ours where the curl of \(\mathbf{F}\) is zero.

Line Integrals

Line integrals are a way to measure a vector field along a path or curve. They are a direct sum of vectors along a line, offering a measure of the field's influence on that path. For this exercise, following the parametrization given, the ellipse described at the shell's bottom boundary is crucial. The equation is \(\mathbf{r}(t) = 3\cos t \, \mathbf{i} + 2\sin t \, \mathbf{j}\), with \(t\) ranging from 0 to \(2\pi\).
Stokes' Theorem allows us to equate the surface integral of a curl on a surface to a line integral over its boundary. For our vector field \(\mathbf{F}\), Stokes' simplifies our efforts since we calculated that the curl \(abla \times \mathbf{F}\) was zero. Typically, these line integrals provide insight into work done or energy transferred along the path, essential in physics and engineering applications.