Problem 18 - Calculus Early Transcendentals

Question

Determine whether the statement is true or false. Explain your answer. (In Exercises 16–18, assume that C is a simple, smooth, closed curve, oriented counterclockwise.) It must be the case that $$ \int_{C} e^{x^{2}} d x+\sin y^{3} d y=0 $$

Step-by-Step Solution

Answer

The statement is true because the conditions of Green's Theorem are satisfied.

Step 1: Understand the Statement and Curve Description

We are given a closed curve \( C \) that is simple, smooth, and oriented counterclockwise. The exercise asks us to evaluate if the line integral along \( C \) of the function \( e^{x^2} \, dx + \sin(y^3) \, dy \) is equal to zero.

Step 2: Recognize the Condition for Green's Theorem

For Green's theorem to apply, which equates a line integral around a closed curve to a double integral over the region it encloses, the vector field must have continuous partial derivatives and must be path independent, implying a curl \( (\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}) \) of zero.

Step 3: Evaluate the Curl of the Given Vector Field

Identify \( M(x,y) = e^{x^2} \) and \( N(x,y) = \sin(y^3) \).Calculate the partial derivatives: \( \frac{\partial N}{\partial x} = 0 \) and \( \frac{\partial M}{\partial y} = 0 \). This implies the curl \( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} = 0 - 0 = 0 \), suggesting the conditions for Green's theorem are satisfied.

Step 4: Apply Green's Theorem Conclusion

Green's Theorem states that with a zero curl, the line integral around the closed curve \( C \) is zero. This supports the statement \( \int_{C} e^{x^{2}} \, dx + \sin(y^3) \, dy = 0 \).

Step 5: Determine the Truth Value of the Statement

Since the necessary conditions of Green’s theorem are satisfied with the curl being zero, the evaluation of the path integral indeed results in zero. Therefore, the statement is true.

Key Concepts

Line Integrals

Line integrals are a fundamental concept in calculus, particularly when dealing with vector fields. They represent the integral of a function along a curve, essentially summing up a quantity over a path. In our exercise, we have the line integral \( \int_{C} e^{x^2} \, dx + \sin(y^3) \, dy = 0 \) to evaluate.

The concept can be thought of in terms of physical interpretation, where the curve \( C \) might represent a path, and the function being integrated represents some physical quantity like force or work.

• The first component \( e^{x^2} \, dx \) sums changes in the \( x \)-direction, weighted by \( e^{x^2} \).
• The second component \( \sin(y^3) \, dy \) sums changes in the \( y \)-direction, weighted by \( \sin(y^3) \).

When the path \( C \) is closed and Green's Theorem conditions are satisfied, the line integral evaluates to zero. This is because the changes over the closed path "cancel out" due to symmetry and the properties of the vector field involved.

Vector Fields

A vector field assigns a vector to every point in the plane. For our exercise, the vector field is composed of functions \( M(x,y) = e^{x^2} \) and \( N(x,y) = \sin(y^3) \).

Understanding vector fields is crucial as they allow us to visualize how quantities like force fields or fluid flow behave in a region.

• Each vector attached to a point indicates direction and magnitude, describing how the field changes at that location.
• The field's behavior guides us in applying Green's Theorem, which relates the path integral of a vector field around a closed curve to the double integral over the region bounded by the curve.

In essence, assessing the vector field helps determine potential characteristics like curl or divergence, which are key to verifying conditions required for the theorems used in line integrals.

Path Independence

Path independence is an essential property that emerges in the context of conservative fields. A line integral being zero around a closed path indicates path independence.

This suggests the integral's value only depends on the endpoints, not the specific path taken between them.

• When a vector field is conservative, the line integral is path-independent, and one can define a potential function.
• For Green’s Theorem to apply—and to confirm path independence—the curl of the vector field must be zero.

In the exercise, since the curl, \( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} = 0 \), is zero, it confirms path independence. Thus, the integral over \( C \) indeed results in zero, verifying the statement given in the problem.