Problem 17 - Advanced Engineering Mathematics

Question

We have \(\oint_{C} x d z=\int_{C_{1}} x d z+\int_{C_{2}} x d z+\int_{C_{3}} x d z\) On \(C_{1}, y=0,0 \leq x \leq 1, z=x, d z=d x\), \(\int_{C_{1}} x d z=\int_{0}^{1} x d x=\frac{1}{2}\). On \(C_{2}, x=1,0 \leq y \leq 1, z=1+i y, d z=i d y\), \(\int_{C_{2}} x d z=i \int_{0}^{1} d y=i\). On \(C_{3}, y=x, 0 \leq x \leq 1, z=x+i x, d z=(1+i) d x\), \(\int_{C_{3}} x d z=(1+i) \int_{1}^{0} x d x=-\frac{1}{2}-\frac{1}{2} i\). Thus \(\oint_{C} x d z=\frac{1}{2}+i-\frac{1}{2}-\frac{1}{2} i=\frac{1}{2} i\).

Step-by-Step Solution

Answer

The result of \( \oint_C x \, dz \) is \( \frac{1}{2} i \).

Step 1: Understanding the Path Integrals

The given integrals represent the path integrals of the function \( x \) over three different curves: \( C_1 \), \( C_2 \), and \( C_3 \). Each curve is described with specific conditions for \( x \), \( y \), and \( z \). We need to evaluate each path integral separately and then sum them to find \( \oint_{C} x \, dz \).

Step 2: Analyzing Path \( C_1 \)

On curve \( C_1 \), \( y = 0 \), \( 0 \leq x \leq 1 \), and \( z = x \). Since \( dz = dx \), the integral becomes \( \int_{C_1} x \, dz = \int_0^1 x \, dx = \frac{1}{2} \). This calculation uses the basic formula for the area under \( x \) from 0 to 1.

Step 3: Analyzing Path \( C_2 \)

On curve \( C_2 \), \( x = 1 \), \( 0 \leq y \leq 1 \), and \( z = 1 + i y \). The derivative is \( dz = i \, dy \), leading to the integral \( \int_{C_2} x \, dz = i \int_0^1 1 \, dy = i \). Here we integrate a constant \( x = 1 \) along \( y \).

Step 4: Analyzing Path \( C_3 \)

On curve \( C_3 \), \( y = x \), \( 0 \leq x \leq 1 \), and \( z = x + i x \). The derivative is \( dz = (1+i) \, dx \). The integral becomes \( \int_{C_3} x \, dz = (1+i) \int_0^1 x \, dx = -\frac{1}{2} - \frac{1}{2} i \), where we compute the integral and account for reversing limits.

Step 5: Summing the Integrals

Combining all individual path integral results: \( \oint_{C} x \, dz = \int_{C_1} x \, dz + \int_{C_2} x \, dz + \int_{C_3} x \, dz = \frac{1}{2} + i - \frac{1}{2} - \frac{1}{2} i = \frac{1}{2} i \). We add the results of each path to find the total value of the closed contour integral.

Key Concepts

Path Integrals

Path integrals are a fundamental idea in calculus, especially within the context of complex analysis. They involve integrating a function over a curve or path. This means we aren't just integrating in a straight line but follow a curve through space. Think of it like walking along a winding path and adding up some value (like altitude) along the way. In complex analysis, these paths are often in the complex plane, with complex functions.
To understand path integrals better, consider a path defined by a parametric curve. The integral calculates the accumulation of some quantity, evaluated along this path.

• One common example is finding the work done by a force along a path.
• In mathematics, we often deal with functions defined by complex variables.

The goal is often to compute the integral over each part of the path separately and then combine these results.

Contour Integrals

Contour integrals are a specific type of path integrals used primarily in complex analysis. They are used to integrate over contours, which are curves in the complex plane. A contour integral takes a complex-valued function and integrates it along a path, or contour, in the complex plane.
Contour integrals are employed for several purposes:

• Evaluating complex functions along specific paths.
• Solving problems related to complex-valued functions, especially in physical applications like fluid dynamics and electromagnetism.

To compute a contour integral, you must know the curve or path you're integrating over, usually represented as a parametric equation. Each segment of the contour may have its own conditions, making the calculations diverse and interesting.

Complex Analysis

Complex analysis is the field of mathematics that studies functions that operate on complex numbers. It's a powerful branch of mathematics, with applications in many areas like engineering, physics, and signal processing.
Key components of complex analysis involve:

• Complex numbers, which include both real and imaginary parts.
• Functions that are defined using these numbers.
• Interesting properties such as differentiability and integrability in the complex plane.

In particular, complex analysis makes extensive use of path and contour integrals. These techniques allow mathematicians to solve complex equations and prove powerful theorems, such as Cauchy's integral theorem and residue theorem, which are pivotal in integrating complex functions.

Integration along Curves

Integration along curves means evaluating integrals over paths that are not necessarily straight. In the context of complex line integrals, these curves often reside in the complex plane and require special rules and methods to compute.
There are several key aspects of integrating along curves:

• Choosing your parameterization: Each curve needs a clear description, often involving parameters like time or angle to describe points along the curve.
• Differential elements: These help in expressing small segments of the curve, crucial for setting up the integral.
• Appraising the function along the path described by these parameters.

Integrating along curves is a generalization of the idea of integration from basic calculus, expanding the scope to more complex paths and functions. This technique is not just limited to theoretical mathematics but is also used in practical applications such as calculating the flow of fluids along specific pathways or electrical potentials in circuits.