Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-6, use Stokes's Theorem to calculate $$ \iint_{S}(\operatorname{curl} \mathbf{F}) \cdot \mathbf{n} d S $$ \(\mathbf{F}=x^{2} \mathbf{i}+y^{2} \mathbf{j}+z^{2} \mathbf{k} ; \quad S\) is the hemisphere \(z=\) \(\sqrt{1-x^{2}-y^{2}}\) and \(\mathbf{n}\) is the upper normal.

Step-by-Step Solution

Answer

The integral is 0.

Step 1: Calculate Curl of F

First, we need to calculate the curl of the vector field \( \mathbf{F} = x^2 \mathbf{i} + y^2 \mathbf{j} + z^2 \mathbf{k} \). Recall that the curl is given by \( abla \times \mathbf{F} = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z} \right) \mathbf{i} + \left( \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x} \right) \mathbf{j} + \left( \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \mathbf{k} \)."

Step 2: Apply Partial Derivatives

1. \( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z} = \frac{\partial z^2}{\partial y} - \frac{\partial y^2}{\partial z} = 0 \)2. \( \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x} = \frac{\partial x^2}{\partial z} - \frac{\partial z^2}{\partial x} = 0 \)3. \( \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} = \frac{\partial y^2}{\partial x} - \frac{\partial x^2}{\partial y} = 0 \)Thus, \( abla \times \mathbf{F} = \mathbf{0} \). This implies that curl \( \mathbf{F} \) is zero.

Step 3: Evaluate the Surface Integral

We need to evaluate \( \iint_{S}(\operatorname{curl} \mathbf{F}) \cdot \mathbf{n} \, dS \). Since \( \operatorname{curl} \mathbf{F} = \mathbf{0} \), we have \( \iint_{S}(\operatorname{curl} \mathbf{F}) \cdot \mathbf{n} \, dS = \iint_{S}0 \cdot \mathbf{n} \, dS = 0 \).

Step 4: Use Stokes's Theorem

Stokes's Theorem states that \( \iint_{S}(\operatorname{curl} \mathbf{F}) \cdot \mathbf{n} \, dS = \oint_{C}\mathbf{F} \cdot d\mathbf{r} \), where \( C \) is the boundary of \( S \). For the hemisphere, the boundary is the circle \( x^2 + y^2 = 1, z = 0 \). However, since the initial integral is zero, \( \oint_{C}\mathbf{F} \cdot d\mathbf{r} = 0 \).

Key Concepts

Curl of a Vector Field

The curl of a vector field is a fundamental concept in vector calculus. It measures the rotation or swirling of a field at a given point. Whenever you encounter a vector field \( \mathbf{F} \), such as \( x^2 \mathbf{i} + y^2 \mathbf{j} + z^2 \mathbf{k} \), calculating the curl helps to determine if the field has any rotational tendencies.

To compute the curl of \( \mathbf{F} \), use the formula:\[abla \times \mathbf{F} = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z} \right) \mathbf{i} + \left( \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x} \right) \mathbf{j} + \left( \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \mathbf{k}\]Here, each term involves partial derivatives. Applying this to our example yields zero for each component. Thus, the curl is \(\mathbf{0}\), meaning there's no tangible rotation in the field.

Surface Integral

The surface integral in this context is the integral of the curl of the vector field over a given surface \( S \). In many physics and engineering applications, this represents the flow or flux of the field through a surface.

When \( \operatorname{curl} \mathbf{F} = 0 \), the surface integral also evaluates to zero because you are integrating a zero vector over the surface.

• The expression \( \iint_{S} (\operatorname{curl} \mathbf{F}) \cdot \mathbf{n} \, dS \) entails taking the dot product of the curl with the normal vector \( \mathbf{n} \).
• Since the curl is zero, it clearly follows that the surface integral of \( \mathbf{F} \) over any surface \( S \) results in zero.

This means that there is no net circulation or flow through the surface \( S \), a characteristic that's significant when applying Stokes's Theorem.

Partial Derivatives

Partial derivatives are essential when working with vector fields because they provide a way to examine how a vector field changes in a specific direction.

When calculating the curl, you take partial derivatives of the components of the vector field. This means differentiating individual pieces of the vector field concerning one variable at a time, while treating all other variables as constants.

• The process entails computing differences between mixed partial derivatives such as:\( \frac{\partial x^2}{\partial y} \) and \( \frac{\partial y^2}{\partial x} \), both of which equate to zero in our example.
• Such computations tell us how each direction in the vector field might give rise to rotational aspects.

Understanding partial differentiation is integral for analyzing complex fields in multi-dimensional space.

Boundary of a Surface

In Stokes's Theorem, the boundary of a surface \( S \), denoted as \( C \), plays a crucial role. This is the path or loop that marks the edge of the surface.

Stokes's Theorem relates a surface integral over \( S \) to a line integral over its boundary \( C \). For the hemisphere in our example, the boundary is the circle defined by \( x^2 + y^2 = 1, z = 0 \).

• The boundary often serves as a key element in applying the theorem, as the integral over the boundary gives information about circulation around the surface.
• When \( \operatorname{curl} \mathbf{F} = 0 \), it indicates that \( \oint_{C} \mathbf{F} \cdot d\mathbf{r} = 0 \) as well.

Understanding how the boundary affects calculations helps in visualizing and comprehending the interplay between line and surface integrals in the theorem.