Problem 12 - Advanced Engineering Mathematics

Question

Use Stokes' theorem to evaluate \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\). Assume \(C\) is oriented counterclockwise as viewed from above. \(\mathbf{F}=z \mathbf{i}+x \mathbf{j}+y \mathbf{k} ; \boldsymbol{C}\) the curve of intersection of the plane \(x+y+z=0\) and the sphere \(x^{2}+y^{2}+z^{2}=1[\) Hint \(:\) Recall that the area of an ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1\) is \(\pi a b\). \(]\)

Step-by-Step Solution

Answer

The integral evaluates to \( \pi \frac{\sqrt{3}}{3} \).

Step 1: Understanding Stokes' Theorem

Stokes' Theorem relates a surface integral of the curl of a vector field to a line integral around its boundary. It is expressed as: \[ \oint_{C} \mathbf{F} \cdot d \mathbf{r} = \iint_{S} (abla \times \mathbf{F}) \cdot d\mathbf{S} \] where \( abla \times \mathbf{F} \) is the curl of \( \mathbf{F} \), and \( S \) is a surface with boundary \( C \).

Step 2: Compute the Curl of F

Calculate the curl of the vector field \( \mathbf{F} = z \mathbf{i} + x \mathbf{j} + y \mathbf{k} \). Using the curl formula, \( abla \times \mathbf{F} = ( \frac{\partial k}{\partial y} - \frac{\partial j}{\partial z} ) \mathbf{i} + ( \frac{\partial i}{\partial z} - \frac{\partial k}{\partial x} ) \mathbf{j} + ( \frac{\partial j}{\partial x} - \frac{\partial i}{\partial y} ) \mathbf{k} \), we get \[ abla \times \mathbf{F} = \mathbf{i} + \mathbf{j} + \mathbf{k} \].

Step 3: Determine the Surface S

The curve \( C \) is where the plane \( x+y+z=0 \) intersects the sphere \( x^2+y^2+z^2=1 \), forming an elliptical loop. Let's use the plane as the surface \( S \), and the normal vector for this plane is \( \mathbf{n} = \langle 1, 1, 1 \rangle \).

Step 4: Evaluate the Surface Integral

To evaluate \( \iint_{S} (abla \times \mathbf{F}) \cdot d\mathbf{S} \), calculate \( (abla \times \mathbf{F}) \cdot \mathbf{n} \). We have \( \mathbf{n} = \frac{1}{\sqrt{3}}(\mathbf{i} + \mathbf{j} + \mathbf{k}) \) for unit normal. Thus, \( (abla \times \mathbf{F}) \cdot \mathbf{n} = 3/\sqrt{3} = \sqrt{3} \).

Step 5: Calculate the Area of the Elliptical Surface

The elliptical region inside the sphere on the plane \( x+y+z=0 \) can be parameterized as an ellipse with semi-axes scaled by \( \sqrt{3}/3 \) due to \( x+y+z=0 \), projecting onto the plane. The area is given by \( \pi \frac{\sqrt{3}}{3} \frac{\sqrt{3}}{3} = \pi \frac{1}{3} \).

Step 6: Compute the Integral

Finally, apply Stokes' Theorem: \[ \iint_{S} (abla \times \mathbf{F}) \cdot d\mathbf{S} = \sqrt{3} \times \pi \frac{1}{3} = \pi \frac{\sqrt{3}}{3} \]. The value of the line integral \( \oint_{C} \mathbf{F} \cdot d \mathbf{r} \) is equal to this result.

Key Concepts

Surface Integrals

Surface integrals allow us to calculate the flow of a vector field across a surface. They can be thought of as an extension of line integrals, where we integrate over a two-dimensional surface instead of a one-dimensional curve. In a surface integral, we sum up the effects of a vector field over every small bit of the surface.
A surface integral of a vector field \( \mathbf{F} \) over a surface \( S \) can be expressed as:

• \( \iint_{S} \mathbf{F} \cdot d\mathbf{S} \)

Here, \( d\mathbf{S} \) is a small area element on the surface that is oriented according to the surface's normal vector.
This integral computes the total vector flux through the surface, considering both the magnitude and direction of the vector field at each point on the surface. In the context of Stokes' Theorem, the surface integral is crucial because it connects the curl of the vector field to the circulation along the boundary.

Vector Fields

Vector fields are mathematical representations indicating the influence, direction, and magnitude of some force or property over a region. They are represented by functions that assign a vector to every point in space.
Imagine a field of arrows over a plane, where each arrow points in the direction of the field at that location.
A vector field \( \mathbf{F} \) in three-dimensional space is commonly expressed as:

• \( \mathbf{F}(x, y, z) = f(x, y, z) \mathbf{i} + g(x, y, z) \mathbf{j} + h(x, y, z) \mathbf{k} \)

Where \( f, g, \) and \( h \) are functions representing the components of the field in \( x, y, \) and \( z \) directions, respectively.
Understanding vector fields is essential as they model real-world phenomena such as electromagnetic fields, gravitational fields, and velocity fields in fluid dynamics.

Curl of a Vector Field

The curl of a vector field measures the tendency of particles to rotate about a point, acting somewhat like a rotation detector for the field. It provides a vector field that represents the axis of the rotating motion and its magnitude.
Mathematically, the curl of a vector field \( \mathbf{F} = f \mathbf{i} + g \mathbf{j} + h \mathbf{k} \) is expressed by:

• \( abla \times \mathbf{F} = ( \frac{\partial h}{\partial y} - \frac{\partial g}{\partial z} ) \mathbf{i} + ( \frac{\partial f}{\partial z} - \frac{\partial h}{\partial x} ) \mathbf{j} + ( \frac{\partial g}{\partial x} - \frac{\partial f}{\partial y} ) \mathbf{k} \)

This operation yields another vector field representing the circulation density at any given point.
In the context of Stokes' Theorem, the curl helps relate the surface integral of a vector field to its line integral around a boundary curve.

Elliptical Area Calculation

The calculation of an elliptical area is something needed when using Stokes' Theorem in scenarios involving elliptical regions. When a planar section intersects a sphere, the intersection often forms an ellipse.
The standard equation of an ellipse is:

• \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)

Here, \( a \) and \( b \) are the semi-major and semi-minor axes, respectively, and the area \( A \) of the ellipse is calculated as \( \pi a b \).
In practice, such as the given problem, it's crucial to determine the ellipse properties resulting from the plane-sphere intersection to compute the surface integral correctly. Recognizing when such calculations are needed is a significant part of applying mathematical theorems to geometric contexts.