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Chapter 9: Problem 2

Find the value of \(f(r, \theta, \phi)=r^{2} \sin ^{2} \theta \cos ^{2} \phi+2 \cos ^{2} \theta-r^{3} \sin 2 \theta \sin \phi\) for (i) \((r, \theta, \phi)=(1, \pi / 2,0)\) (ii) \((r, \theta, \phi)=(2, \pi / 4, \pi / 6)\) (iii) \((r, \theta, \phi)=(0, \pi, \pi / 3)\).

Short Answer

Expert verified
(i) 1, (ii) -1.5, (iii) 2.

Step by step solution

01

Substitute values for (i)

For the coordinates \((r, \theta, \phi) = (1, \pi/2, 0)\):Substitute these into the function:\[f(1, \pi/2, 0) = 1^2 \sin^2 (\pi/2) \cos^2 (0) + 2\cos^2 (\pi/2) - 1^3 \sin (2 \cdot \pi/2) \sin (0)\] Since \(\sin(\pi/2) = 1\) and \(\cos(0) = 1\):\[f(1, \pi/2, 0) = 1^2 \cdot 1^2 \cdot 1^2 + 2 \cdot 0^2 - 1^3 \cdot 0 \cdot 0 = 1 + 0 - 0 = 1\].
02

Substitute values for (ii)

For the coordinates \((r, \theta, \phi) = (2, \pi/4, \pi/6)\):Substitute these into the function:\[f(2, \pi/4, \pi/6) = 2^2 \sin^2 (\pi/4) \cos^2 (\pi/6) + 2 \cos^2 (\pi/4) - 2^3 \sin (2 \cdot \pi/4) \sin (\pi/6)\]Calculate:\(\sin(\pi/4) = \frac{\sqrt{2}}{2}\), \(\cos(\pi/6) = \frac{\sqrt{3}}{2}\), and \(\sin(2 \cdot \pi/4) = \sin(\pi/2) = 1\).Thus:\[f(2, \pi/4, \pi/6) = 4 \cdot \left(\frac{\sqrt{2}}{2}\right)^2 \cdot \left(\frac{\sqrt{3}}{2}\right)^2 + 2 \cdot \left(\frac{\sqrt{2}}{2}\right)^2 - 8 \cdot 1 \cdot \frac{1}{2}\]Which simplifies to:\[4 \cdot \frac{1}{2} \cdot \frac{3}{4} + 2 \cdot \frac{1}{2} - 4 = 1.5 + 1 - 4 = -1.5\].
03

Substitute values for (iii)

For the coordinates \((r, \theta, \phi) = (0, \pi, \pi/3)\):Substitute these into the function:\[f(0, \pi, \pi/3) = 0^2 \sin^2 (\pi) \cos^2 (\pi/3) + 2 \cos^2 (\pi) - 0^3 \sin (2 \cdot \pi) \sin (\pi/3)\]Since \(\sin(\pi) = 0\) and \(\cos(\pi) = -1\):\[f(0, \pi, \pi/3) = 0 + 2 \cdot 1 - 0 = 2\].

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Key Concepts

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

Trigonometric Functions
Trigonometric functions play a crucial role in solving problems involving spherical coordinates and polar equations. Key trigonometric functions include sine (\(\sin\)) and cosine (\(\cos\)), which relate the angles of a right triangle to the lengths of its sides. They are fundamental to understanding coordinate transformations.

In this exercise, various angles specified in radians (\(\pi/2\), \(\pi/4\), etc.) are used to evaluate the function \(f(r, \theta, \phi)\). Important trigonometric values such as \(\sin(\pi/2)=1\) and \(\cos(0)=1\) simplify the expressions involved in evaluating the function, reducing complexity in the calculations.

Key points:
  • Understanding how to evaluate sine and cosine for standard angles is essential.
  • Trigonometric functions help compute relationships involving polar and spherical coordinates.
  • These functions capture the geometry of rotations and angles in different coordinate systems.
Mathematical Problem Solving
Mathematical problem-solving often involves substituting given values into expressions or equations to find solutions. The systematic approach to solving such problems involves careful substitution, calculation, and simplification.

In this exercise:
  • Each step begins with substituting the given values of \((r, \theta, \phi)\) into the function \(f(r, \theta, \phi)\)
  • Simplification of trigonometric functions using known values such as \(\sin(\pi/4)=\frac{\sqrt{2}}{2}\) or \(\cos(\pi/6)=\frac{\sqrt{3}}{2}\)
  • Breaking down complex problems into smaller, manageable parts makes solving them more straightforward.
Approaching each sub-problem with clarity and systematically performing calculations ensures accuracy and helps build mathematical intuition.
Polar Equations
Polar equations are mathematical expressions where points on a plane are defined using polar coordinates (\((r, \theta)\)). They differ from traditional Cartesian coordinates (\((x, y)\)) by using a radius and an angle. This system is particularly useful for modeling circular patterns and periodic motions.

In the context of spherical coordinates, which add a third angle \(\phi\), we extend polar equations into three-dimensional space. This allows for representation and calculation on spheres, an important concept in physics and engineering.

Key considerations:
  • Understand the difference between polar and Cartesian coordinates, focusing on how angles affect positioning.
  • Recognize how \(\theta\) and \(\phi\) alter the point's placement on a spherical surface.
  • Use these concepts to interpret how spherical equations can represent 3D space, aiding in the solution of real-world engineering or physics problems.

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Most popular questions from this chapter

Given \(f=\sin (u+v)\), where \(v=\cos u\), (i) find \(\frac{d f}{d u}\), (ii) if \(u=e^{-t}\), find \(\frac{d f}{d t}\).

Given \(z=x^{2}+2 x y+3 y^{2}\), where \(x=(1+t)^{1 / 2}\) and \(y=(1-t)^{1 / 2}\), find \(\frac{d z}{d t}\) by (i) substitution, (ii) the chain rule (9.21).

Evaluate the integral and sketch the region of integration: $$ \int_{0}^{a} \int_{0}^{\sqrt{a^{2}-x^{2}}} x y^{2} d y d x $$

Find the stationary points of the following functions: $$ 3-x^{2}-x y-y^{2}+2 y $$

Find all the first and second partial derivatives of $$ z=\sin (x+y) e^{x-y} $$
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