Problem 11 - 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. $$ \begin{aligned} &\mathbf{F}=x \mathbf{i}+x^{3} y^{2} \mathbf{j}+z \mathbf{k} ; C \text { the boundary of the semi-ellipsoid }\\\ &z=\sqrt{4-4 x^{2}-y^{2}} \text { in the plane } z=0 \end{aligned} $$

Step-by-Step Solution

Answer

The integral evaluates to zero.

Step 1: Understand Stokes' Theorem

Stokes' Theorem states that \( \oint_{C} \mathbf{F} \cdot d\mathbf{r} = \iint_{S} (abla \times \mathbf{F}) \cdot \mathbf{n} \, dS \), where \( S \) is a surface bounded by the contour \( C \) and \( \mathbf{n} \) is the unit normal to \( S \). We need to apply this theorem to find the integral over the closed curve \( C \).

Step 2: Compute the Curl of \( \mathbf{F} \)

First, determine the curl of \( \mathbf{F} \), which is given by \( abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \partial x & \partial y & \partial z \ x & x^3 y^2 & z \end{vmatrix} \). Calculate each component to get \( abla \times \mathbf{F} = (0 - 2x^3 y) \mathbf{i} - (1 - 0) \mathbf{j} + (0 - 0) \mathbf{k} = -2x^3 y \mathbf{i} - \mathbf{j} \).

Step 3: Define the Surface \( S \)

The boundary \( C \) is the projection of the semi-ellipsoid onto the \( xy \)-plane. The surface \( S \) can be the upper half of this semi-ellipsoid, defined by \( z = \sqrt{4 - 4x^2 - y^2} \) with \( z \geq 0 \). The unit normal vector for the upward orientation is derived from the gradient of \( f(x, y, z) = 4 - 4x^2 - y^2 - z^2 \), yielding \( \mathbf{n} = \left( \frac{4x}{z}, \frac{y}{z}, 1 \right) \).

Step 4: Evaluate the Surface Integral

The surface integral is \( \iint_{S} (-2x^3y, -1, 0) \cdot \left( \frac{4x}{z}, \frac{y}{z}, 1 \right) \, dS \). This becomes \( \iint_{D} \left( -\frac{8x^4y}{z} - \frac{y}{z} \right) \sqrt{1 + \left( \frac{\partial z}{\partial x} \right)^2 + \left( \frac{\partial z}{\partial y} \right)^2} \, dx \, dy \) over the projected region \( D \) in the \( xy \)-plane, where \(z = \sqrt{4 - 4x^2 - y^2} \).

Step 5: Simplify and Evaluate the Double Integral

Substitute \( z = \sqrt{4 - 4x^2 - y^2} \) into the integral and simplify using the symmetry of the region. The computation is complex, so directly compute in cylindrical or polar coordinates or perform a numerical integration if necessary. Simplifications using symmetry reveal that the integral evaluates to zero due to cancellations in the symmetry of the function and region.

Key Concepts

Vector Calculus

Vector calculus is a field of mathematics that deals with vector fields and operations on them. It extends traditional calculus to multiple dimensions and is essential in understanding the behavior of physical quantities in space. In vector calculus, you work with vector fields, which are functions that assign a vector to each point in space. These vectors can represent anything from the velocity of a fluid to the force per unit mass in a gravitational field.

Key concepts include:

• Gradient: It provides the slope or rate of change of a scalar field and points in the direction of the greatest rate of increase of the function.
• Divergence: It measures how much a vector field spreads out from a given point. It is important in fields like fluid dynamics and electromagnetism.
• Curl: As explored in this problem with Stokes' Theorem, curl measures the rotation or twist of a vector field around a point.

Understanding these concepts is vital for applying formulas like Stokes' Theorem. Stokes' Theorem connects surface integrals with line integrals through the curl of a vector field, which is a perfect example of how vector calculus is used to solve complex problems in physics and engineering.

Surface Integrals

Surface integrals extend the concept of integration from lines and planes to complex surfaces. In essence, they allow you to add up a quantity over a surface in three-dimensional space. Surface integrals are crucial for calculating net flux across a surface, among many other applications.

When performing a surface integral, you first need to parametrize the surface, often using two variables such as \(u\) and \(v\). This converts the surface into a form that can be integrated over in two dimensions rather than three. For example, the surface defined as \(z = \sqrt{4 - 4x^2 - y^2}\) in the problem statement translates into a half-ellipsoid.

Steps to evaluate a surface integral include:

• Identifying and parametrize the surface
• Calculating the differential element of area \(dS\)
• Determining the unit normal vector, which is often obtained by normalizing the cross-product of the partial derivatives of the parametrization
• Substituting these into the integral to evaluate

These steps help simplify complex surfaces into manageable calculations, especially when using the tools provided by vector calculus, like Stokes' Theorem.

Curl of a Vector Field

Curl is a fundamental concept in vector calculus that provides crucial insights into the rotation of a vector field. For a given vector field \(\mathbf{F}\), the curl helps determine how the field "swirls" around any given point. It is a vector operation that results in another vector, showing the axis of rotation and magnitude of rotational strength.

To compute the curl of a vector field \(\mathbf{F} = (F_1, F_2, F_3)\), we use the determinant involving partial derivatives:\[abla \times \mathbf{F} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} F_1 & F_2 & F_3 \end{vmatrix}\]

In the given problem, the vector field \(\mathbf{F}=x \mathbf{i}+x^{3} y^{2} \mathbf{j}+z \mathbf{k}\) results in the curl \(abla \times \mathbf{F} = -2x^3 y \hat{\mathbf{i}} - \hat{\mathbf{j}} + 0 \hat{\mathbf{k}}\). This indicates the nature of the field's rotation around any point. The zero coefficient in \(\hat{\mathbf{k}}\) shows no rotational component in that direction.

Understanding curl is essential for applying Stokes' Theorem effectively. The theorem makes use of curl to relate surface integrals with line integrals, showing the deep connection between geometry and calculus in vector fields.