Question

Special line integrals Prove the following identities, where \(C\) is a simple closed smooth oriented curve. \(\int_{C} f(x) d x+g(y) d y=0,\) where \(f\) and \(g\) have continuous derivatives on the region enclosed by \(C\)

Short answer

Question: Prove that the integral \(\int_{C} f(x) d x + g(y) d y = 0\) for a simple closed smooth oriented curve \(C\) and two functions \(f(x)\) and \(g(y)\) with continuous derivatives on the region enclosed by \(C\). Answer: Using Green's theorem and showing that the partial derivatives of \(f(x)\) and \(g(y)\) with respect to \(x\) and \(y\) are both zero, we proved that the integral \(\int_{C} f(x) d x + g(y) d y = 0\).

Step-by-step solution

Recall Green's theorem

Green's theorem states that if \(\textbf{F} = P(x, y) \textbf{i} + Q(x, y) \textbf{j}\) is a vector field with continuous first partial derivatives on an open region containing \(D\) where \(C\) is a simple, closed, and positively-oriented curve bounding region \(D\), then: \(\oint_{C} (P d x + Q d y) = \iint_{D} (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) dA.\) In our case, we have the vector field \(\textbf{F}(x, y) = f(x) \textbf{i} + g(y) \textbf{j}\). In order to satisfy the conditions of Green's theorem, we will first compute the partial derivatives of \(f(x)\) and \(g(y)\) to obtain the necessary form.

Calculate the partial derivatives of \(f(x)\) and \(g(y)\)

Since \(\textbf{F}(x, y) = f(x) \textbf{i} + g(y) \textbf{j}\), we have \(P(x, y) = f(x)\) and \(Q(x, y) = g(y)\). We need to compute the partial derivatives \(\frac{\partial P}{\partial y}\) and \(\frac{\partial Q}{\partial x}\). \(\frac{\partial P}{\partial y} = \frac{\partial f(x)}{\partial y} = 0\) because \(f(x)\) does not depend on \(y\). Similarly, \(\frac{\partial Q}{\partial x} = \frac{\partial g(y)}{\partial x} = 0\) because \(g(y)\) does not depend on \(x\).

Apply Green's theorem

Now that we know \(\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x} = 0\), we can apply Green's theorem as follows: \(\oint_{C} (f(x) d x + g(y) d y) = \iint_{D} (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) dA = \iint_{D} (0 - 0) dA = 0\). Therefore, according to Green's theorem, the integral \(\oint_{C} f(x) d x + g(y) d y = 0\). And we have proven the given identity.

Key concepts

Line Integrals

Line integrals are a fascinating way to extend the concept of integration to curves. Instead of finding the area under a curve, we evaluate a function along a curve. This curve can be in two or three dimensions. Think of it as walking along a path and adding up values of a function at different points on the path. Line integrals are used to evaluate the work done by a force field, among other applications. In mathematical terms, a line integral of a vector field \( \textbf{F}(x, y) = P(x, y) \textbf{i} + Q(x, y) \textbf{j} \) along a curve \( C \) is given by:\[\int_{C} P \, dx + Q \, dy\]In our problem, the curve \( C \) is closed, meaning it returns to its starting point, and smooth, meaning there are no sharp turns. When applying line integrals, it's crucial to consider the orientation this path takes, as direction can impact the result.

Vector Fields

Vector fields are like imaginary maps where each point is associated with a vector (a direction and a magnitude). These fields are very common in physics where they represent quantities like velocity fields in fluids or electromagnetic fields.A vector field is usually described by a function, \( \textbf{F}(x, y) = P(x, y) \textbf{i} + Q(x, y) \textbf{j} \), where \( P \) and \( Q \) are functions of \( x \) and \( y \). In our given problem, the vector field is simple: \( \textbf{F}(x, y) = f(x) \textbf{i} + g(y) \textbf{j} \). Note how this field only depends on one variable at a time for each component, which impacts how we go about solving line integrals with it. Understanding how the vector field flows and behaves is vital when applying the line integrals since the field's nature will determine if Green’s Theorem can be applied, as it was in the solution.

Partial Derivatives

Partial derivatives are like regular derivatives but for functions with multiple variables. They show how a function changes as one of the variables changes while the other remains constant. In the context of vector fields and Green’s Theorem, partial derivatives play a critical role. For example, for the function \( P(x, y) = f(x) \), the partial derivative with respect to \( y \) is \( \frac{\partial f(x)}{\partial y} = 0 \) because \( f(x) \) does not depend on \( y \). Similarly, for \( Q(x, y) = g(y) \), the partial derivative with respect to \( x \) is \( \frac{\partial g(y)}{\partial x} = 0 \).These calculations are essential in applying Green's theorem accurately. The idea is to find the difference between these partial derivatives, which in our problem becomes zero, allowing us to conclude that the integral of the vector field over the closed curve \( C \) is zero.