Problem 30 - Advanced Engineering Mathematics

Question

A vector space \(V\) on which a dot or inner product has been defined is called an inner product space. An inner product for the vector space \(C[a, b]\) is given by $$ (f, g)=\int_{a}^{b} f(x) g(x) d x $$ In \(C[0,2 \pi]\) compute \((x, \sin x)\).

Step-by-Step Solution

Answer

The value of \((x, \sin x)\) over \([0, 2\pi]\) is \(2\pi\).

Step 1: Identify the Function Components

To compute the inner product \((x, \sin x)\) in \(C[0,2\pi]\), we identify the functions involved. Here, the functions are \(f(x) = x\) and \(g(x) = \sin x\).

Step 2: Set Up the Inner Product Integral

The inner product \((x, \sin x)\) is defined by the integral \(\int_{0}^{2\pi} x \sin x \; dx\). This integral represents the dot product of the functions \(f(x)\) and \(g(x)\) over the interval \([0, 2\pi]\).

Step 3: Integration by Parts Formula

Use the integration by parts formula to evaluate the integral. The formula is \(\int u \, dv = uv - \int v \, du\). Here, choose \(u = x\) and \(dv = \sin x \, dx\).

Step 4: Differentiate and Integrate Components

Differentiate \(u = x\) to get \(du = dx\). Integrate \(dv = \sin x \, dx\) to get \(v = -\cos x\).

Step 5: Apply Integration by Parts

Substitute into the integration by parts formula: \(\int x \sin x \, dx = -x \cos x \big|_0^{2\pi} + \int \cos x \, dx\).

Step 6: Evaluate the Boundary Terms

Evaluate \(-x \cos x\) at the boundaries: \[-x \cos x \big|_0^{2\pi} = [-(2\pi)(-1) - 0] = 2\pi\].

Step 7: Solve Remaining Integral

Solve \(\int \cos x \, dx\), which becomes \(\sin x\).Evaluate at boundaries: \(\sin x \big|_0^{2\pi} = \sin(2\pi) - \sin(0) = 0 - 0 = 0\).

Step 8: Compute the Final Result

Combine these results to get the final answer: \(\int_{0}^{2\pi} x \sin x \, dx = 2\pi + 0 = 2\pi\).

Key Concepts

Vector Space

A vector space is a mathematical structure formed by a collection of objects called vectors. These vectors can be added together and multiplied by scalars, to produce another vector in the same space. This concept is foundational in mathematics and physics because it helps in understanding linear transformations and matrix operations.

Key Attributes of a Vector Space:

• Vectors in a vector space can be scaled and added, conserving the vector space properties.
• The zero vector, acting as an additive identity, always exists.
• Every vector has an additive inverse, ensuring that for any vector, there is another that cancels it out.

In the context of inner product spaces, these vector spaces have additional structure which involves the inner or dot product. The inner product provides a way to define angles and lengths, bringing geometric interpretation into play.

C[0,2π]

The notation \(C[0, 2\pi]\) represents the space of continuous functions defined on the closed interval \([0, 2\pi]\). These functions are continuous and infinite dimensional, making \(C[0, 2\pi]\) a classic example of a function space in analysis.

Features of \(C[0, 2\pi]\):

• Includes all functions that are continuous on the interval from 0 to \(2\pi\).
• Common examples of functions in this space include polynomial and trigonometric functions.
• Integral operations, like finding inner products, are well defined due to the continuity of functions.

The vector space \(C[0, 2\pi]\) is significant when discussing inner product spaces, as it provides a natural setting for applying mathematical tools like the dot product in the context of integrals.

Integration by Parts

Integration by parts is a powerful technique used to solve integrals in calculus. It is particularly useful for integrals involving a product of two functions, where direct integration is not straightforward. The method is derived from the product rule of differentiation.

Integration by Parts Formula:
The formula is given by \[\int u \, dv = uv - \int v \, du\]This formula allows for a specific choice of \(u\) and \(dv\), enabling easier integration.

Steps to Apply:

• Identify parts of the function to assign to \(u\) and \(dv\).
• Differentiate \(u\) to find \(du\), and integrate \(dv\) to find \(v\).
• Substitute into the integration by parts formula to simplify the integral.

In the example of integral \(\int x \sin x \, dx\), the choice was \(u = x\) and \(dv = \sin x \, dx\). This specific assignment simplifies the integration process, making it more manageable.

Dot Product

The dot product, also known as the inner product, is a critical operation in vector algebra. It combines two vectors to produce a scalar. The dot product helps determine orthogonality and angles between vectors.

Properties of Dot Product:

• The dot product is commutative; that is, \(\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}\).
• If the dot product of two vectors is zero, they are orthogonal (perpendicular).
• In vector spaces of functions, the dot product is often defined via integrals, as seen in \(C[0, 2\pi]\).

In function spaces such as \(C[0, 2\pi]\), the dot product is expressed as \[(f, g) = \int_{a}^{b} f(x) g(x) \, dx\]This operation is crucial for analyzing properties like orthogonality and norms in the context of functions.