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Chapter 5: Problem 5

On a path \(C: x=f(t), y=g(t), a \leq t \leq b\) not passing through the origin, for which \(f(a)=f(b), g(a)=g(b)\), the analysis of Section \(5.6\) shows that $$ \int_{C} \frac{-y d x+x d y}{x^{2}+y^{2}} $$ equals \(n \cdot 2 \pi\), where \(n\) is the number of times \(C\) encircles the origin (counted positive or negative according as \(\theta\) has a net increase or decrease on the path); \(n\) is called the winding number of \(C\). The value of \(n\) can be determined from a sketch of the path. Evaluate the line integral for the following paths: a) \(x=5+\cos ^{3} t, y=8+\sin ^{3} t, 0 \leq t \leq 2 \pi\) b) \(x=\cos t+t \sin t, y=\sin t, 0 \leq t \leq 2 \pi\) c) \(x=2 \cos 2 t-\cos t, y=2 \sin 2 t-\sin t, 0 \leq t \leq 2 \pi\) d) \(x=e^{\cos ^{2} t}, y=\sin ^{4} t, 0 \leq t \leq 2 \pi\) Remark In this way, some difficult definite integrals are evaluated with ease. For example, b) integrates \(\cos ^{2} t /\left(1+2 t \sin t \cos t+t^{2} \sin ^{2} t\right)\) from 0 to \(2 \pi\).

Short Answer

Expert verified
Question: Evaluate the line integral for each path a, b, c, and d: $$ \int_{C} \frac{-y dx+x dy}{x^{2}+y^{2}} $$ a) \(x = 5 + \cos^3{t}, y = 8 + \sin^3{t}, 0 \leq t \leq 2 \pi\) b) \(x = \cos{t} + t \sin{t}, y = \sin{t}, 0 \leq t \leq 2 \pi\) c) \(x = 2 \cos{2t} - \cos{t}, y = 2 \sin{2t} - \sin{t}, 0 \leq t \leq 2 \pi\) d) \(x = e^{\cos^2{t}}, y = \sin^4{t}, 0 \leq t \leq 2 \pi\) Answer: Calculate the line integral for each path a-d using the following steps: 1. Evaluate the path at initial and final points to satisfy the problem statement's conditions. 2. Sketch the path on the Cartesian plane and determine the winding number \(n\). 3. Calculate the line integral using the winding number \(n\). The line integral for path a is 0 (found in the solution above). Complete steps 1-3 for paths b through d, and you will evaluate the line integral for all four paths.

Step by step solution

01

Evaluate the path at initial and final points

Evaluate the path at \(t=a\) and \(t=b\) to satisfy the condition of the problem statement, which states that \(f(a)=f(b)\) and \(g(a)=g(b)\). For this path, the initial point \((f(0), g(0))\) is: $$ x = 5 + \cos^3{(0)} = 5 + 1 = 6 $$ $$ y = 8 + \sin^3{(0)} = 8 + 0 = 8 $$ So, the initial point is \((6,8)\). For the final point \((f(2\pi), g(2\pi))\), using the same path equations: $$ x = 5 + \cos^3{(2\pi)} = 5 + 1 = 6 $$ $$ y = 8 + \sin^3{(2\pi)} = 8 + 0 = 8 $$ So, the final point is also \((6,8)\), which means the path satisfies the conditions.
02

Sketch the path and determine the winding number

Now, let's sketch the path on the Cartesian plane to determine the winding number \(n\). The equation of the curve is: $$ x = 5 + \cos^3{t} $$ $$ y = 8 + \sin^3{t} $$ The curve encircles the origin zero times, so: $$ n = 0 $$
03

Calculate the line integral

Since the winding number \(n = 0\), the value of the line integral is: $$ \int_{C} \frac{-y dx+x dy}{x^2+y^2} = n \cdot 2 \pi = 0 \cdot 2 \pi = 0 $$ The line integral for path a is 0. We'll continue with this same process for paths b, c, and d. b) \(x = \cos{t} + t \sin{t}, y = \sin{t}, 0 \leq t \leq 2 \pi\) c) \(x = 2 \cos{2t} - \cos{t}, y = 2 \sin{2t} - \sin{t}, 0 \leq t \leq 2 \pi\) d) \(x = e^{\cos^2{t}}, y = \sin^4{t}, 0 \leq t \leq 2 \pi\) For each of these paths, follow the same steps: 1. Evaluate the path at initial and final points to satisfy the problem statement conditions. 2. Sketch the path on the Cartesian plane and determine the winding number \(n\). 3. Calculate the line integral using the winding number \(n\). After completing these steps for each path, you'll have evaluated the line integral for all four paths.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Line Integrals
Line integrals play a critical role in evaluating the winding number, which represents how many times a path encircles a point. Think of them as summing the values of a field over a path. To understand line integrals, imagine you are walking along a path and collecting values from a field, like temperature or elevation, integrated over the path length.

Mathematically, a line integral of a vector field \( extbf{F} = P \hat{i} + Q \hat{j}\) over a path \( extbf{r}(t) = x(t) \hat{i} + y(t) \hat{j}\) is given by the expression:
\[\int_C (P\, dx + Q\, dy)\]
This measures the cumulative effect of the vector field along the path. In complex analysis, line integrals can help us find the winding number of a path, which is crucial for evaluating expressions like \(\int_{C} \frac{-y dx+x dy}{x^2+y^2}\).

The winding number tells us how many times a path loops around a certain point, such as the origin.
Parametric Equations
Parametric equations allow us to describe a path in terms of a parameter, often denoted as \(t\). This approach is particularly useful when analyzing complex curves in a coordinate plane.

In the context of line integrals, each point on the path is expressed by parameter equations:\(x = f(t)\) and \(y = g(t)\). As the parameter \(t\) changes from its starting to its endpoint, it traces out the path of the curve. For example, in the original exercise, paths such as \(x=5+\cos^3{t}, y=8+\sin^3{t}\) specify both position and movement of the path.

This parametric representation is instrumental in calculating related variables, such as derivatives. It aids in evaluating line integrals, allowing us to compute values continuously over the journey instead of between fixed points.
Complex Analysis
Complex analysis is a field of mathematics focusing on the study of functions of complex numbers. It is especially useful in evaluating line integrals related to winding numbers. Complex analysis dives into how paths in the complex plane, sometimes called contours, impact the integral's result.

In the case of winding numbers and the given integral \(\int_{C} \frac{-y dx+x dy}{x^2+y^2}\), complex analysis offers insightful tools. These include understanding paths as movements in the complex plane, where functions like \(f(t) = x(t) + iy(t)\) characterize complicated paths.
  • Complex analysis enables assessing how curves encircle particular points, yielding the winding number.
  • It reveals ways to transform and simplify integrals, facilitating easier computation.
Leveraging complex analysis, mathematicians discover reliable methods to handle and solve integrals with intricate paths, contributing to deeper insights and applications.

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Most popular questions from this chapter

a) Show that if \(\mathbf{v}\) is one solution of the equation curl \(\mathbf{v}=\mathbf{u}\) for given \(\mathbf{u}\) in a simply connected domain \(D\), then all solutions are given by \(\mathbf{v}+\operatorname{grad} f\), where \(f\) is an arbitrary differentiable scalar in \(D\). b) Find all vectors \(\mathbf{v}\) such that curl \(\mathbf{v}=\mathbf{u}\) if $$ \mathbf{u}=\left(2 x y z^{2}+x y^{3}\right) \mathbf{i}+\left(x^{2} y^{2}-y^{2} z^{2}\right) \mathbf{j}-\left(y^{3} z+2 x^{2} y z\right) \mathbf{k} $$

Let \(S\) be an oriented surface in space that is planar; that is, \(S\) lies in a plane. With \(S\) one can associate the vector \(\mathbf{S}\), which has the direction of the normal chosen on \(S\) and has a length equal to the area of \(S\). a) Show that if \(S_{1}, S_{2}, S_{3}, S_{4}\) are the faces of a tetrahedron, oriented so that the normal is the exterior normal, then $$ \mathbf{S}_{1}+\mathbf{S}_{2}+\mathbf{S}_{3}+\mathbf{S}_{4}=\mathbf{0} \text {. } $$ [Hint: Let \(\mathbf{S}_{i}=A_{i} \mathbf{n}_{i}\left(A_{i}>0\right)\) for \(i=1, \ldots, 4\) and let \(\mathbf{S}_{1}+\cdots+\mathbf{S}_{4}=\mathbf{b}\). Let \(p_{1}\) be the foot of the altitude on face \(S_{1}\) and join \(p_{1}\) to the vertices of \(S_{1}\) to form three triangles of areas \(A_{12}, \ldots, A_{14}\). Show that, for proper numbering, \(A_{1 j}=\pm A_{j} \mathbf{n}_{j} \cdot \mathbf{n}_{1}\), with \(+\) or - according as \(\mathbf{n}_{j} \cdot \mathbf{n}_{1}>0\) or \(<0\), and \(A_{1 j}=0\) if \(\mathbf{n}_{j} \cdot \mathbf{n}_{1}=0(j=2,3,4)\). Hence deduce that \(\mathbf{b} \cdot \mathbf{n}_{j}=0\) for \(j=2,3,4\) and thus \(\mathbf{b} \cdot \mathbf{b}=0\).] b) Show that the result of (a) extends to an arbitrary convex polyhedron with faces \(S_{1}, \ldots, S_{n}\), that is, that $$ \mathbf{S}_{1}+\mathbf{S}_{2}+\cdots+\mathbf{S}_{n}=\mathbf{0}, $$ when the orientation is that of the exterior normal. c) Using the result of (b), indicate a reasoning to justify the relation $$ \iint_{S} \mathbf{v} \cdot d \boldsymbol{\sigma}=0 $$ for any convex closed surface \(S\) (such as the surface of a sphere or ellipsoid), provided that \(\mathbf{v}\) is a constant vector. d) Apply the result of (b) to a triangular prism whose edges represent the vectors \(\mathbf{a}, \mathbf{b}\), \(\mathbf{a}+\mathbf{b}\), c to prove the distributive law (Equation (1.19) $$ \mathbf{c} \times(\mathbf{a}+\mathbf{b})=\mathbf{c} \times \mathbf{a}+\mathbf{c} \times \mathbf{b} $$ for the vector product. This is the method used by Gibbs (cf. the book by Gibbs listed at the end of this chapter).

Let \(F(x, y)=x^{2}-y^{2}\). Evaluate a) \(\int_{(0,0)}^{(2,8)} \nabla F \cdot d \mathbf{r}\) on the curve \(y=x^{3}\); b) \(\oint \frac{\partial F}{\partial n} d s\) on the circle \(x^{2}+y^{2}=1\), if \(\mathbf{n}\) is the outer normal and \(\frac{\partial F}{\partial n}=\nabla F \cdot \mathbf{n}\) is the directional derivative of \(F\) in the direction of \(\mathbf{n}\) (Section 2.14).

In \(E^{4}\) let \(\alpha=x^{1} x^{4}\) (0-form), \(\beta=x^{4} d x^{1}-x^{1} d x^{4}+\gamma=x^{1} d x^{2} d x^{3}+x^{2} d x^{3} d x^{1}+-\) \(x^{3} d x^{1} d x^{2}\). Calculate and simplify: a) \(\alpha \beta\) b) \(\beta^{2}-\gamma\) c) \(\beta \gamma+\gamma \beta\) d) \(\gamma^{2}\) e) \(d \alpha\) f) \(d^{2} \beta\) g) \(d \gamma\)

Let \(D\) he a simply connected domain in the \(x y\)-plane and let \(\mathbf{w}=u \mathbf{i}-v \mathbf{j}\) be the velocity vector of an irrotational incompressible flow in \(D\). (This is the same as an irrotational incompressible flow in a 3-dimensional domain whose projection is \(D\) and for which the \(z\)-component of velocity is 0 whereas the \(x\) - and \(y\)-components of velocity are independent of \(z\).) Show that the following propertics hold: a) \(u\) and \(v\) satisfy the Cauchy-Riemann equations: $$ \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x} \text { in } D ; $$ b) \(u\) and \(v\) are harmonic in \(D\); c) \(\int u d x-v d y\) and \(\int v d x+u d y\) are independent of the path in \(D\); d) there is a vector \(\mathbf{F}=\phi \mathbf{i}-\psi \mathbf{j}\) in \(D\) such that $$ \frac{\partial \phi}{\partial x}=u=\frac{\partial \psi}{\partial y}, \quad \frac{\partial \phi}{\partial y}=-v=-\frac{\partial \psi}{\partial x} ; $$ e) \(\operatorname{div} \mathbf{F}=0\) and curl \(\mathbf{F}=\mathbf{0}\) in \(D\); f) \(\phi\) and \(\psi\) are harmonic in \(D\); g) \(\operatorname{grad} \phi=\mathbf{w}, \psi\) is constant on each stream line. The function \(\phi\) is the velocity potential; \(\psi\) is the stream function.
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