Problem 332 - Calculus Volume 3

Question

For the following exercises, use Stokes' theorem to evaluate \(\iint_{S}(\operatorname{curl} \mathbf{F} \cdot \mathbf{N}) d S\) for the vector fields and surface. \(\mathbf{F}(x, y, z)=x y \mathbf{i}-z \mathbf{j}\) and \(S\) is the surface of the cube \(0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1,\) except for the face where \(z=0,\) and using the outward unit normal vector.

Step-by-Step Solution

Answer

The integral evaluates to -2 over the specified surface.

Step 1: Understand Stokes' Theorem

Stokes' theorem relates the surface integral of the curl of a vector field over a surface \( S \) to the line integral of the vector field over the boundary curve \( C \) of \( S \). It is given by: \( \iint_{S} (\operatorname{curl} \mathbf{F} \cdot \mathbf{N}) \, dS = \oint_{C} \mathbf{F} \cdot \, d\mathbf{r} \). In this problem, we're tasked to apply Stokes' theorem where \( S \) is the given surface.

Step 2: Determine the boundary of the surface

The surface \( S \) is the boundary of a cube missing the face at \( z=0 \). The boundary \( C \) therefore consists of edges not lying in the \( z=0 \) plane, i.e., forming a closed loop along the edges where \( z=1 \), \( x=0 \), \( x=1 \), \( y=0 \), and \( y=1 \) only excluding the face \( z=0 \).

Step 3: Calculate \( \operatorname{curl} \mathbf{F} \)

The given vector field is \( \mathbf{F}(x, y, z) = xy \mathbf{i} - z \mathbf{j} \). We find \( \operatorname{curl} \mathbf{F} = abla \times \mathbf{F} \). The curl in three dimensions is calculated as:\[abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ xy & -z & 0 \end{vmatrix}\]This results in \( \operatorname{curl} \mathbf{F} = \mathbf{k} \times (x) + \mathbf{j} \times (y) + \mathbf{i} \times (-1) \), which simplifies to \( \operatorname{curl} \mathbf{F} = y \mathbf{k} + \mathbf{j} \cdot 0 + (-1) \mathbf{i} = -\mathbf{i} + y \mathbf{k} \).

Step 4: Identify the Line Integral

Using Stokes’ theorem, \( \oint_{C} \mathbf{F} \cdot \, d\mathbf{r} \) accounts for this integral around the path at the top \( z=1 \): going from \( (0,0,1) \) to \( (1,0,1) \), then to \( (1,1,1) \), next to \( (0,1,1) \), and finally back to \( (0,0,1) \). We need to evaluate \( \mathbf{F} \cdot \, d\mathbf{r} \) along each segment of this loop.

Step 5: Evaluate the Line Integral on each Segment

The path on \( z=1 \) involves four line integral segments, calculate each:- **Segment 1:** Parametrize \( x \) from \( 0 \) to \( 1 \), \( y=0, z=1 \), i.e. \( d\mathbf{r} = dx \, \mathbf{i} \).- **Segment 2:** \( x = 1 \), \( y \) from \( 0 \) to \( 1 \), \( z=1 \), i.e. \( d\mathbf{r} = dy \, \mathbf{j} \).- **Segment 3:** \( x \) from \( 1 \) to \( 0 \), \( y = 1 \), \( z=1 \), i.e. \( d\mathbf{r} = -dx \, \mathbf{i} \).- **Segment 4:** \( x = 0 \), \( y \) from \( 1 \) to \( 0 \), \( z=1 \), i.e. \( d\mathbf{r} = -dy \, \mathbf{j} \).

Step 6: Simplify and Calculate Each Segment

For each segment, evaluate \( \mathbf{F} \cdot \, d\mathbf{r} \):- **Segment 1:** \( \int_{0}^{1} xy\, dx = 0 \), as \( y = 0 \). - **Segment 2:** \( \int_{0}^{1} (-z) \, dy = -1 \), with \( z=1 \).- **Segment 3:** \( \int_{1}^{0} xy \, (-dx) = 0 \), as \( y = 1 \). - **Segment 4:** \( \int_{1}^{0} (-z) \, (-dy) = -1 \).

Step 7: Sum Results of all Segments

The complete integral is the sum of the results from each segment.\( 0 - 1 + 0 - 1 = -2 \). Based on symmetry and continuation, adjustments to sign and path enforce this sum as the consistent outcome.

Key Concepts

Surface Integral

Understanding the surface integral is crucial when applying Stokes' Theorem. It is a form of integral calculus that allows us to understand how a vector field interacts over a surface. In the context of Stokes' Theorem, the surface integral involves calculating \( \iint_{S} (\operatorname{curl} \mathbf{F} \cdot \mathbf{N}) \, dS \), where:

• \( \operatorname{curl} \mathbf{F} \) represents the curl of the vector field \( \mathbf{F} \).
• \( \mathbf{N} \) is the unit normal vector to the surface \( S \).
• \( dS \) denotes an infinitesimal piece of the surface.

The surface integral aggregates the effects of the vector field curling around tiny patches within a surface, allowing us to derive the total effect over the entire surface. By understanding how a field behaves over \( S \), we can infer the presence of rotations or swirls due to the field's vector nature.

Line Integral

Line integrals are integral calculations over a curve or path rather than a surface. In Stokes’ Theorem, it connects the concept of line integrals to that of surface integrals. The theorem states:\[ \iint_{S} (\operatorname{curl} \mathbf{F} \cdot \mathbf{N}) \, dS = \oint_{C} \mathbf{F} \cdot \, d\mathbf{r} \]Here,

• \( \oint_{C} \mathbf{F} \cdot \, d\mathbf{r} \) is the line integral over the boundary \( C \) of the surface \( S \).
• \( C \) is typically a closed path or loop.
• The dot product \( \cdot \) signifies that the vector field \( \mathbf{F} \) is multiplied by the tangent vector \( d\mathbf{r} \) at each point on the path.

Thus, the line integral measures the tendency of the field \( \mathbf{F} \) to align with or oppose the path \( C \). Here, we evaluate it on the boundary formed by edges of the cube, excluding the face where \( z=0 \). This helps in computing circulation around the loop.

Curl of a Vector Field

The curl of a vector field is an essential concept in vector calculus. It measures the rotation or swirling strength of a field around a point. Calculating the curl involves taking the cross product of the del operator \( abla \) and the vector field \( \mathbf{F} \). This is expressed in three dimensions as:\[ \operatorname{curl} \mathbf{F} = abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ F_x & F_y & F_z \end{vmatrix} \]Where:

• \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) are the standard unit vectors in the x, y, and z directions.
• The partial derivatives \( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \) represent the rate of change of the components of \( \mathbf{F} \).

In the problem, \( \mathbf{F}(x, y, z) = xy \mathbf{i} - z \mathbf{j} \), calculating the curl gives \( \operatorname{curl} \mathbf{F} = -\mathbf{i} + y \mathbf{k} \). This vector indicates how the field circulates in space around \( S \).

Parametrization of a Curve

Parametrization of a curve is a method to represent the path over which a line integral is taken in terms of one or more parameters. It transforms the curve into a path that can be easily manipulated and integrated. For the boundary of surface \( S \) in the context of Stokes' Theorem, we used parametrization to handle each segment of the path \( C \), like the top surface of the cube where \( z=1 \). The parametrization rules:

• Segment 1: A straight line from \( (0,0,1) \) to \( (1,0,1) \), where you vary \( x \), keeping \( y \) and \( z \) constant.
• Segment 2: A vertical line from \( (1,0,1) \) to \( (1,1,1) \), increasing \( y \).
• Segment 3: A horizontal movement back to \( (0,1,1) \), decreasing \( x \).
• Segment 4: Back down vertically to \( (0,0,1) \), decreasing \( y \).

Using parametrization, the path can be sliced into manageable pieces for evaluating \( \oint_{C} \mathbf{F} \cdot \, d\mathbf{r} \). This step is vital for partitioning a complex path into simpler, calculable segments.