Question

Special line integrals Prove the following identities, where \(C\) is a simple closed smooth oriented curve. $$\oint_{C} d x=\oint d y=0$$

Short answer

Question: Prove that the line integrals of \(dx\) and \(dy\) over a simple closed smooth curve \(C\) are both zero. Answer: By applying Green's theorem and selecting appropriate vector fields, we demonstrated that both line integrals \(\oint_{C} dx\) and \(\oint_{C} dy\) are equal to zero for a simple closed smooth curve \(C\). Specifically, for \(\oint_{C} dx\), we used the vector field \(\mathbf{F}(x, y) = (1, 0)\), and for \(\oint_{C} dy\), we used the vector field \(\mathbf{F}(x, y) = (0, 1)\). Thus, the given identities hold: \(\oint_{C} dx =\oint_{C} dy = 0\).

Step-by-step solution

Recall Green's theorem

Green's theorem states that for a positively oriented, simple, smooth closed curve \(C\) in the plane, and a continuously differentiable vector field \(\mathbf{F}(x, y) = (P(x, y), Q(x, y))\), the following relation holds: $$\oint_{C} \mathbf{F} \cdot d\mathbf{r} = \iint_{D} \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA$$ where \(D\) is the region bounded by \(C\), and \(d\mathbf{r} = (dx, dy)\). In our problem, we have to compute \(\oint_{C} dx\) and \(\oint_{C} dy\).

Computing the line integral of \(dx\)

To compute \(\oint_{C} dx\), we consider the vector field \(\mathbf{F}(x, y) = (1, 0)\). Notice that the partial derivatives are: $$\frac{\partial Q}{\partial x} = 0, \qquad \frac{\partial P}{\partial y} = 0$$ Now we apply Green's theorem: $$\oint_{C} dx = \oint_{C} \mathbf{F} \cdot d\mathbf{r} = \iint_{D} \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA = \iint_{D} (0-0) dA = 0$$

Computing the line integral of \(dy\)

To compute \(\oint_{C} dy\), we consider the vector field \(\mathbf{F}(x, y) = (0, 1)\). Again, we compute the partial derivatives: $$\frac{\partial Q}{\partial x} = 0, \qquad \frac{\partial P}{\partial y} = 0$$ We apply Green's theorem once more: $$\oint_{C} dy = \oint_{C} \mathbf{F} \cdot d\mathbf{r} = \iint_{D} \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA = \iint_{D} (0-0) dA = 0$$

Conclusion

We have shown that both line integrals \(\oint_{C} dx\) and \(\oint_{C} dy\) are equal to zero for a simple closed smooth curve \(C\). Therefore, the given identities hold: $$\oint_{C} dx =\oint_{C} dy = 0$$

Key concepts

Green's theorem

Green's Theorem is a powerful tool in calculus, connecting the concept of a line integral around a closed curve with a double integral over the region enclosed by the curve. Using this theorem, we can convert a difficult line integral into a more manageable area integral. It specifically relates the circulation of a vector field around a closed curve to the sum of the partial derivatives inside the enclosed area. The formal expression of Green's theorem is:
\[\oint_{C} \mathbf{F} \cdot d\mathbf{r} = \iint_{D} \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA\]
Here, \( \mathbf{F}(x, y) \) is the vector field, \( C \) is the positively oriented smooth closed curve, and \( D \) is the region enclosed by \( C \). Understanding and applying Green's theorem requires familiarity with vector fields, line integrals, partial derivatives, and the orientation of curves.

Closed curve line integrals

Line integrals along closed curves play a significant role in fields such as physics and engineering, where they're often used to compute work done by a force field along a path. A closed curve line integral, denoted by the integral symbol with a circle, \(\oint_C\), signifies integration over a path that returns to its starting point without crossing itself. When we compute \(\oint_C dx\) or \(\oint_C dy\), we're summing up the components of the vector field along the tangent to the curve. However, when this is done over a simple closed curve with no net flow across the boundary, the total sum becomes zero. This principle is crucial for understanding conservation laws and the behavior of vector fields along boundaries.

Vector field integration

Integrating a vector field along a curve gives us the net effect of the field along that path. When dealing with closed curves, this can indicate something about the nature of the field within the bounded region. Vector field integration, similar to regular integration, accumulates the field's influence along a path or over an area. When we talk about line integrals of vector fields, we are often interested in their work or circulation, which captures the field's tendency to move a particle along the curve. For example, in physics, this is related to concepts like electric potential and fluid flow. A zero line integral, as shown in our textbook exercise solution, can indicate a sort of equilibrium within the enclosed area—no net circulation or work done.

Partial derivative applications

Partial derivatives tell us how a function behaves as we vary one of its variables while keeping the others constant. They are foundational in the study of multivariable calculus and have a myriad of applications. In the context of Green's theorem, we use partial derivatives to express how a vector field changes in each direction—these changes can tell us about sources, sinks, or circulation within a vector field. Whenever Green's theorem is invoked, we're essentially examining these minute changes across an area, allowing us to deduce information about larger-scale behaviors from localized behaviors. This fundamental application of partial derivatives helps us connect local properties of a vector field to global results, as demonstrated in the original textbook problem.