Problem 16 - 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.) If $$ \int_{C} f(x, y) d x+g(x, y) d y=0 $$ then \(\partial g / \partial x=\partial f / \partial y\) at all points in the region bounded by \(C .\)

Step-by-Step Solution

Answer

False; the integral being zero doesn't imply equality at each point.

Step 1: Understanding the Green's Theorem

Green's Theorem relates a line integral around a simple closed curve \(C\) to a double integral over the plane region \(R\) bounded by \(C\). It states: \[ \int_{C} (P\, dx + Q\, dy) = \iint_R \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dA \].

Step 2: Applying Green's Theorem to the Given Integral

Given the integral \(\int_{C} f(x,y)\, dx + g(x,y)\, dy = 0 \), identify that \(P = f(x,y)\) and \(Q = g(x,y)\). According to Green's Theorem, \(\iint_R \left( \frac{\partial g}{\partial x} - \frac{\partial f}{\partial y} \right) \, dA = 0 \).

Step 3: Analyzing the Result of the Double Integral

Since \(\iint_R \left( \frac{\partial g}{\partial x} - \frac{\partial f}{\partial y} \right) \, dA = 0 \), it means that the expression under the double integral must integrate to zero over the entire region \(R\).

Step 4: Determining Conditions for Zero Double Integral

The integral of a function over a region being zero doesn't necessarily mean the function itself is zero everywhere within the region. It could mean the function averages to zero over \(R\). In some cases, the function can be positive in parts of \(R\) and negative in others, balancing to zero.

Step 5: Conclude the Truth of the Statement

The given statement claims that \( \frac{\partial g}{\partial x} = \frac{\partial f}{\partial y} \) at all points in \(R\). However, the result of the integral only ensures that the average rate of change over \(R\) is zero, not at every individual point. Thus, the statement is false.

Key Concepts

Line Integral

A line integral is a type of integral where a function is evaluated along a curve. Think of it as measuring how much a function accumulates along a path. This is useful in many fields such as physics and engineering where you calculate the work done by a force field along a path or curve.
In mathematical terms, a line integral of a vector field \( \vec{F} = P \hat{i} + Q \hat{j} \) along a curve \( C \) is written as \( \int_{C} P \, dx + Q \, dy \). Here, \( P \) and \( Q \) are functions of \( x \) and \( y \) respectively.

• When you compute a line integral around a closed curve, you're essentially summing up values that describe the net flow or circulation of a vector field along that curve.
• In the context of Green's Theorem, line integrals relate to the circulation around a closed boundary.

Understanding line integrals is crucial, especially when you wish to apply the theoretical implications of Green's Theorem, connecting the work done around a curve to a function's properties in the region it encloses.

Closed Curve

A closed curve in mathematics is a path that starts and ends at the same point without crossing or intersecting itself. This forms a boundary around a particular region in the plane.

• In the problem context, the curve \( C \) is specifically defined to be smooth and simple, meaning it doesn’t have any sharp turns or points where it crosses itself.
• Understanding closed curves is vital in applying Green's Theorem, as it specifically looks at the behavior of functions around these loops.
• Additionally, a closed curve ensures that the region enclosed can be filled or defined fully within a plane, useful for understanding integrations over an area.

For applications such as in Green's Theorem, a closed curve provides the perimeter through which a line integral is computed, connecting the curve and the area it encloses.

Partial Derivatives

Partial derivatives are derivatives of functions with more than one variable where each variable is treated independently. They measure the rate at which a function changes as one of the variables changes, keeping the others constant.

• The notation \( \frac{\partial f}{\partial x} \) denotes the partial derivative of a function \( f(x, y) \) with respect to \( x \), while treating \( y \) as a constant parameter.
• In the context of Green's Theorem and this problem, partial derivatives such as \( \frac{\partial g}{\partial x} \) and \( \frac{\partial f}{\partial y} \) are significant because they determine the differences in rates of change across a bounded region.
• The key aspect in this exercise is understanding how these derivatives relate to the net "circulation" across the region and when they might not necessarily equate throughout every point in the region, despite yielding zero in integral form.

Partial derivatives are essential in multivariable calculus, providing insights into how functions behave with each variable independently within the context of larger theorems like Green's.