Problem 13 - Calculus Early Transcendentals: Pearson New International Edition

Question

Suppose that the surface \(S\) is determined by the formula \(z=g(x, y)\). Show that the surface integral in Stokes's Theorem can be written as a double integral in the following way: $$ \iint_{S}(\operatorname{curl} \mathbf{F}) \cdot \mathbf{n} d S=\iint_{S_{x y}}(\operatorname{curl} \mathbf{F}) \cdot\left(-g_{x} \mathbf{i}-g_{y} \mathbf{j}+\mathbf{k}\right) d A $$ where \(\mathbf{n}\) is the upward normal to \(S\) and \(S_{x y}\) is the projection of \(S\) in the \(x y\)-plane.

Step-by-Step Solution

Answer

The surface integral on \( S \) becomes a double integral on its projection, \( S_{xy} \), using the normal vector and projection transformations.

Step 1: Understand the Problem Statement

We are given a surface \( S \) defined by \( z = g(x, y) \) and need to express the surface integral from Stokes's Theorem as a double integral in terms of its projection on the \( xy \)-plane.

Step 2: Identify the Normal Vector

The surface integral in Stokes's Theorem involves the dot product of \( \operatorname{curl} \mathbf{F} \) and the normal vector \( \mathbf{n} \). Since \( S \) is parameterized by \( z = g(x, y) \), an upward normal vector \( \mathbf{n} \) is \( (-g_x \mathbf{i} - g_y \mathbf{j} + \mathbf{k})/\sqrt{g_x^2 + g_y^2 + 1} \).

Step 3: Express dS in Terms of dA

The differential of surface area \( dS \) on \( S \) can be expressed using the projection in the \( xy \)-plane: \( dS = \sqrt{g_x^2 + g_y^2 + 1} \, dA \), where \( dA \) is the area element on the \( xy \)-plane.

Step 4: Substitute Expressions into the Surface Integral

Now, substitute these expressions into the surface integral: \( \iint_S (\operatorname{curl} \mathbf{F}) \cdot \mathbf{n} \, dS = \iint_{S_{xy}} (\operatorname{curl} \mathbf{F}) \cdot (-g_x \mathbf{i} - g_y \mathbf{j} + \mathbf{k}) \, \sqrt{g_x^2 + g_y^2 + 1} \, dA \).

Step 5: Simplify the Expression

Notice that the multiplicative factor \( \sqrt{g_x^2 + g_y^2 + 1} \) cancels out since \( dS = \sqrt{g_x^2 + g_y^2 + 1} \, dA \). The integral simplifies to \( \iint_{S_{xy}} (\operatorname{curl} \mathbf{F}) \cdot (-g_x \mathbf{i} - g_y \mathbf{j} + \mathbf{k}) \, dA \).

Step 6: Conclusion

By substituting and simplifying, we have expressed the surface integral as a double integral over the projection of \( S \) onto the \( xy \)-plane. Thus, \( \iint_S (\operatorname{curl} \mathbf{F}) \cdot \mathbf{n} \, dS = \iint_{S_{xy}} (\operatorname{curl} \mathbf{F}) \cdot (-g_x \mathbf{i} - g_y \mathbf{j} + \mathbf{k}) \, dA \).

Key Concepts

Surface integral

A surface integral is a powerful mathematical tool used to sum up quantities over a curved surface. It is an extension of a double integral, similar to how a line integral extends a single variable integral over a line.

In the context of Stokes's Theorem, a surface integral measures how a vector field, like force or velocity, interacts with a given surface. Specifically, it computes the cumulative effect of field vectors passing through the surface. The formula for a basic surface integral is given by:

• \( \iint_{S} \mathbf{F} \cdot \mathbf{n} \, dS \)

where \( \mathbf{F} \) is the vector field and \( \mathbf{n} \) is the normal vector to the surface \( S \).

Stokes's Theorem connects surface integrals to line integrals, stating that the integral of the curl of a vector field over a surface equals the line integral of the vector field around the boundary of that surface.
Understanding the surface integral in terms of the curl and normal vector, as in the original exercise, helps students grasp this elegant relationship and explore the geometrical and physical behaviors of vector fields.

Curl of a vector field

The curl of a vector field is a concept that measures the tendency of the field to rotate around a given point. It is fundamental in both physics and engineering, as it helps determine the rotational characteristics of the field.

Mathematically, the curl of a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \) in three dimensions is defined as:

• \( abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k} \)

The resulting vector shows the axis and magnitude of rotation in the field.

When applying Stokes's Theorem, the curl is central because it projects the circulating behavior of the vector field onto the surface. Performing a surface integral of the curl, as shown in the exercise, translates this rotational attribute into a comprehensive expression involving a double integral across the surface's projection.

Normal vector

A normal vector is essential in determining the orientation of a surface. It is perpendicular to the surface at a given point and helps define the surface's geometry in three-dimensional space.

For a surface described by \( z = g(x, y) \), the normal vector can be calculated using partial derivatives, which yield:

• \( \mathbf{n} = \left(-g_{x} \mathbf{i} - g_{y} \mathbf{j} + \mathbf{k}\right) / \sqrt{g_{x}^2 + g_{y}^2 + 1} \)

This formula gives an upward-pointing normal vector, indicating it is directed outward from the surface.

The normal vector is crucial in solving surface integrals, as it interacts with other vectors like the curl of the vector field, affecting both the magnitude and direction of the integral. By using the normal vector correctly, the surface integral can be simplified into a double integral, as demonstrated in the exercise, thus bridging complexities across dimensional analyses.

Double integral

A double integral is a method used to compute the area under a surface over a certain region in two-dimensional space. It is especially useful in physics and engineering for calculating properties such as mass, charge, or total energy distributed over an area.

In applying Stokes's Theorem, the surface integral of the curl of a vector field is transformed into a double integral over the projection of the surface onto the \( xy \)-plane. This is represented as:

• \( \iint_{S_{xy}} (abla \times \mathbf{F}) \cdot (-g_x \mathbf{i} - g_y \mathbf{j} + \mathbf{k}) \, dA \)

Such a simplification allows for the analysis to focus on a flatter, two-dimensional equation rather than on the potentially more complex surface itself.

The ability to represent a surface integral as a double integral over a projection simplifies computations and fosters a deeper understanding of the relationship between geometry and vector calculus, making these concepts more approachable for students.