Problem 344 - Calculus Volume 3

Question

Use Stokes' theorem to evaluate line integral \(\int_{C}(z d x+x d y+y d z), \quad\) where \(C\) is a triangle with vertices (3,0,0),(0,0,2) , and (0,6,0) traversed in the given order.

Step-by-Step Solution

Answer

The line integral is zero.

Step 1: Identify the Surface and Boundary

First, identify a surface \( S \) whose boundary is the curve \( C \). The plane that passes through the points (3,0,0), (0,0,2), and (0,6,0) is defined by the equation \( \frac{x}{3} + \frac{z}{2} = 1 \). Thus, the triangular surface \( S \) is bounded by the given vertices and lies within this plane.

Step 2: Parameterize the Surface

Parameterize the surface \( S \) using parameters \( u \) and \( v \) as follows: set \( x = 3u, \, y = 6v, \, z = 2(1-u-v) \). The parameters \( u \) and \( v \) should satisfy \( 0 \leq u \leq 1 \) and \( 0 \leq v \leq 1-u \). This parameterization covers the entire triangular surface.

Step 3: Compute the Surface Normal Vector

The normal vector to the surface can be computed by taking the cross product of the partial derivatives of the parameterization with respect to \( u \) and \( v \). These derivatives are \( \frac{\partial \mathbf{r}}{\partial u} = (3, 0, -2) \) and \( \frac{\partial \mathbf{r}}{\partial v} = (0, 6, -2) \), respectively. The cross product is \( \mathbf{n} = (12, 6, 18) \).

Step 4: Calculate the Curl of the Vector Field

Compute the curl of the vector field \( \vec{F} = (z, x, y) \). The curl is given by \( abla \times \vec{F} = \left(1 - 1, 0 - 0, 0 - 0\right) = (0, 0, 0) \).

Step 5: Evaluate the Surface Integral

Stokes' Theorem relates the surface integral over \( S \) of the curl of \( \vec{F} \) to the line integral over the boundary \( C \). Since \( abla \times \vec{F} = (0, 0, 0) \), the surface integral \( \int_{S} (abla \times \vec{F}) \cdot d\mathbf{S} \) is zero. Thus, the line integral \( \int_{C} (z \; dx + x \; dy + y \; dz) \) is also zero.

Key Concepts

Line Integral

A line integral is a fundamental concept in calculus and vector analysis. It involves integrating a function over a curve. In the context of vector fields, a line integral computes the work done by a field along a path. Imagine you are moving along the curve, and the vector field does work on you; the line integral calculates the total effect.

• The path is usually represented by a vector function \( \mathbf{r}(t) \), where \( t \) is a parameter that varies over an interval.
• In the exercise, the curve \( C \) is a path around a triangle formed by specific points.
• Stokes' Theorem links the computation of a line integral along \( C \) to a surface integral over a surface \( S \) bound by \( C \).

This makes calculations more straightforward when dealing with vector fields. Instead of directly computing the potentially complex line integral, Stokes' Theorem allows us to use the surface integral, which can be simpler to resolve.

Curl of a Vector Field

The curl is an essential operator in vector calculus that measures rotation. Given a vector field \( \vec{F} = (P, Q, R) \), the curl \( abla \times \vec{F} \) captures how much the field "circulates" at a point.

• The mathematical operation of the curl is similar to a cross product and results in a vector.
• In our specific example, the vector field is \( \vec{F} = (z, x, y) \), and the curl \( abla \times \vec{F} \) was found to be \((0, 0, 0)\).
• This zero value indicates that the vector field possesses no rotation in space, simplifying computations like evaluating integrals.

Understanding the curl helps predict how a fluid or field behaves, revealing swirls and eddies. In scenarios involving Stokes' Theorem, a zero curl simplifies evaluating integrals as it indicates no net spinning within the surface.

Surface Integral

Surface integrals extend the idea of line integrals to higher dimensions. They involve integrating a function across a surface. If the surface is open and smooth, it is usually parameterized by two variables, allowing integration over a two-dimensional domain.

• The surface integral accounts for how a field interacts with a surface, considering both the field's direction and the surface's orientation.
• In the context of Stokes' Theorem, the surface \( S \) is the triangular plane bounded by the line integral's curve \( C \).
• After determining the curl of the vector field, you compute the surface integral of this curl over \( S \).
• With the vector field's curl being zero, the surface integral simplifies to zero for our example.

Surface integrals are crucial when applying Stokes' Theorem in assessing how fields and surfaces interact, offering a unified view of circulation-related properties within fields. They're instrumental in physics and engineering, describing how forces act across boundaries.