Problem 15 - Advanced Engineering Mathematics

Question

Use Stokes' theorem to evaluate \(\iint_{S}(\operatorname{curl} \mathbf{F}) \cdot \mathbf{n} d S\). Assume that the surface \(S\) is oriented upward. \(\mathbf{F}=3 x^{2} \mathbf{i}+8 x^{3} y \mathbf{j}+3 x^{2} y \mathbf{k} ; S\) that portion of the plane \(z=x\) that lies inside the rectangular cylinder defined by the planes \(x=0, y=0, x=2, y=2\)

Step-by-Step Solution

Answer

The value of the integral is 88.

Step 1: Understanding Stokes' Theorem

Stokes' theorem relates a surface integral of the curl of a vector field over a surface \( S \) to a line integral over the boundary curve \( C \) of the surface. It is given by \( \iint_{S}(\operatorname{curl} \mathbf{F}) \cdot \mathbf{n} \ dS = \oint_{C} \mathbf{F} \cdot d\mathbf{r} \). We will use this theorem to convert the given surface integral to a line integral.

Step 2: Determine the Boundary Curve

The surface \( S \) is part of the plane \( z = x \) and is bounded by the planes \( x=0, y=0, x=2, y=2 \). The boundary \( C \) is a rectangle lying on the plane \( z=x \) at the edges of the given constraints. The vertices of this rectangle are: \((0,0,0), (2,0,2), (2,2,2), (0,2,0)\).

Step 3: Parametrize the Boundary Curve

To parametrize the boundary curve \( C \), consider each segment of the rectangle: 1) From \( (0,0,0) \) to \( (2,0,2) \), parametrize as \( \mathbf{r}_1(t) = (t, 0, t) \) for \( t \) in \([0, 2]\). 2) From \( (2,0,2) \) to \( (2,2,2) \), parametrize as \( \mathbf{r}_2(u) = (2, u, 2) \) for \( u \) in \([0, 2]\). 3) From \( (2,2,2) \) to \( (0,2,0) \), parametrize as \( \mathbf{r}_3(v) = (2-v, 2, 2-v) \) for \( v \) in \([0, 2]\). 4) From \( (0,2,0) \) to \( (0,0,0) \), parametrize as \( \mathbf{r}_4(w) = (0, 2-w, 0) \) for \( w \) in \([0, 2]\).

Step 4: Calculate \( \mathbf{F} \cdot d\mathbf{r} \) for Each Segment

1) For \( \mathbf{r}_1(t) \), \( d\mathbf{r}_1 = (1, 0, 1) \, dt \) and \( \mathbf{F}(\mathbf{r}_1) = (3t^2, 8t^3 \cdot 0, 3t^2 \cdot 0) = (3t^2, 0, 0) \). Thus, \( \mathbf{F}(\mathbf{r}_1) \cdot d\mathbf{r}_1 = 3t^2 + 0 = 3t^2 \, dt \).2) For \( \mathbf{r}_2(u) \), \( d\mathbf{r}_2 = (0, 1, 0) \, du \) and \( \mathbf{F}(\mathbf{r}_2) = (12u, 64u, 12u) \cdot du \). Thus, \( \mathbf{F}(\mathbf{r}_2) \cdot d\mathbf{r}_2 = 64u \, du \).3) For \( \mathbf{r}_3(v) \), \( d\mathbf{r}_3 = (-1, 0, -1) \, dv \) and \( \mathbf{F}(\mathbf{r}_3) = (3(2-v)^2, 8(2-v)^3 \cdot 2, 3(2-v)^2 \cdot 2) \). Thus, \( \mathbf{F}(\mathbf{r}_3) \cdot d\mathbf{r}_3 = (-3(2-v)^2 - 3(2-v)^2) \cdot dv \).4) For \( \mathbf{r}_4(w) \), \( d\mathbf{r}_4 = (0, -1, 0) \, dw \) and \( \mathbf{F}(\mathbf{r}_4) = (0, 0, 0) \). Thus, \( \mathbf{F}(\mathbf{r}_4) \cdot d\mathbf{r}_4 = 0 \, dw \).

Step 5: Calculate the Line Integral Along the Boundary

Compute \( \oint_{C} \mathbf{F} \cdot d\mathbf{r} = \int_{0}^{2} 3t^2 dt + \int_{0}^{2} 64u du + \int_{0}^{2} (-6(2-v)^2) dv + \int_{0}^{2} 0 \, dw \).The first integral is \( \int_{0}^{2} 3t^2 dt = 3 \left[ \frac{t^3}{3} \right]_0^2 = 8 \).The second integral is \( \int_{0}^{2} 64u du = 64 \left[ \frac{u^2}{2} \right]_0^2 = 128 \).The third integral is \( \int_{0}^{2} (-6(2-v)^2) dv = -6 \left[ \frac{(2-v)^3}{3} \right]_0^2 = -48 \).Add them: \( 8 + 128 - 48 = 88 \).

Step 6: Conclusion

By Stokes' Theorem, the surface integral \( \iint_{S}(\operatorname{curl} \mathbf{F}) \cdot \mathbf{n} \, dS = \oint_{C} \mathbf{F} \cdot d\mathbf{r} = 88 \).

Key Concepts

Surface Integral

In mathematics, a surface integral is used to compute the sum of values over a surface in three dimensions. When dealing with a vector field, we're interested in the flux through a surface. Flux measures how much vector field "flows" through the surface.

To understand surface integrals, imagine a surface divided into tiny patches. You evaluate each patch's contribution by using the field's vector at that patch and the area of the patch. By adding up all these contributions, you obtain the total integral value.

• The vector field: Represents quantities with magnitude and direction over space.
• The surface: Could be flat, like a plane, or curved, like a sphere.
• The integral: Reflects the cumulative effect of the vector field on the surface.

Stokes' theorem simplifies certain surface integrals into easier line integrals along the boundary of the surface, making complex calculations more manageable.

Line Integral

A line integral is a type of integral where a function is evaluated along a curve. Unlike regular integrals, which sum up over intervals, line integrals sum over paths in space.

In the context of a vector field, a line integral helps measure the work done by a field along a curve. Imagine dragging an object along a path in a force field; the line integral gives the total work done.

• Path: Represented typically by a parameterized curve.
• Vector Field: Has components that influence how the curve is traversed.
• Line Integral Result: Could be a scalar, representing work or circulation.

When used with Stokes' theorem, a line integral around a closed path relates to the curl of the field over the surface bounded by that path.

Curl of a Vector Field

The curl of a vector field is a vector operation that describes the rotation or swirling of the field around a point. It's a fundamental concept in vector calculus, often employed in physics and engineering.

To compute the curl, you apply the curl operation—an operator using partial derivatives—to the vector field, resulting in a new vector field.

• Interpretation: It measures local spinning behavior or circulation.
• Computed With: Derivatives, taking into account all vector components.
• Physical Example: The way water swirls around a drain is analogous to a field's curl.

The significance of curl in Stokes' theorem is that it relates the surface integral of curl over a surface to the line integral around the boundary of that surface.

Boundary Curve

A boundary curve is the enclosing path of a surface. It's an essential aspect of surfaces used in Stokes' theorem.

In geometric terms, the boundary curve defines the limit of a surface. If you think of a surface like a sheet of paper, the boundary curve is the paper's edges. Understanding the boundary is crucial because it determines the path for the line integral when using Stokes' theorem.

• Defines: The edge of a surface in 3D space.
• Used in: Calculating line integrals in the context of Stokes' theorem.
• Requires Parametrization: To effectively calculate line integrals.

By parameterizing the boundary curve, one makes it easier to perform integrals and solve problems involving surfaces and their boundaries.