Problem 93 - Precalculus

Question

In calculus, determinants are used when evaluating double and triple integrals through a change of variables. In these cases, the elements of the determinant are functions. Find each determinant. $$\left|\begin{array}{ccc} \sin \phi \cos \theta & -\rho \sin \phi \sin \theta & \rho \cos \phi \cos \theta \\ \sin \phi \sin \theta & \rho \sin \phi \cos \theta & \rho \cos \phi \sin \theta \\ \cos \phi & 0 & -\rho \sin \phi \end{array}\right|$$

Step-by-Step Solution

Answer

The determinant is \( \rho^2 (\sin^3 \phi \cos \theta + \sin \phi \sin \theta (\sin^2 \phi \sin \theta + \cos^2 \phi) - \sin \phi \cos^2 \phi \cos \theta) \).

Step 1: Expand the Determinant

To find the determinant of a 3x3 matrix, use the formula: \[\begin{vmatrix} a & b & c \ d & e & f \ g & h & i \end{vmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg)\]. Here, \(a, b, c, \dots, i\) represent the elements of the matrix. In our matrix: \[a = \sin \phi \cos \theta, \ b = -\rho \sin \phi \sin \theta, \ c = \rho \cos \phi \cos \theta \] \[d = \sin \phi \sin \theta, \ e = \rho \sin \phi \cos \theta, \ f = \rho \cos \phi \sin \theta \] \[g = \cos \phi, \ h = 0, \ i = -\rho \sin \phi \].

Step 2: Calculate the First Term

Calculating the first term from the determinant's expansion: \[ a(ei - fh) \] \[= \sin \phi \cos \theta (\rho \sin \phi (\rho \sin \phi) - \rho \cos \phi \sin \theta \cdot 0) \] \[= \rho \sin \phi \cos \theta (\rho \sin^2 \phi) \] \[= \rho^2 \sin^3 \phi \cos \theta \].

Step 3: Calculate the Second Term

Calculating the second term, which includes a negative sign: \[- b(di - fg) \] \[= -(-\rho \sin \phi \sin \theta) (\sin \phi \sin \theta \cdot (-\rho \sin \phi) - \rho \cos \phi \cos \phi) \] \[= \rho \sin \phi \sin \theta (\rho \sin^2 \phi \sin \theta + \rho \cos^2 \phi) \] \[= \rho^2 \sin \phi \sin \theta (\sin^2 \phi \sin \theta + \cos^2 \phi) \].

Step 4: Calculate the Third Term

The third term is found by evaluating: \[c(dh - eg) \] \[= \rho \cos \phi \cos \theta (\sin \phi \sin \theta \cdot 0 - \rho \sin \phi \cos \phi) \] \[= -\rho^2 \cos \phi \cos \theta \sin \phi \cos \phi \] \[= -\rho^2 \sin \phi \cos^2 \phi \cos \theta \].

Step 5: Combine All Terms

Add all terms calculated in the previous steps to find the determinant: \[\rho^2 \sin^3 \phi \cos \theta + \rho^2 \sin \phi \sin \theta (\sin^2 \phi \sin \theta + \cos^2 \phi) - \rho^2 \sin \phi \cos^2 \phi \cos \theta\]. Simplify further if needed. Since combining might not simplify calculable trigonometric identities effectively, this is kept as is for the determinant form.

Key Concepts

calculus

In calculus, determinants are often used to simplify complex calculations, particularly when dealing with integrals using a change of variables. The determinant is important for evaluating double and triple integrals by helping us transform coordinate systems more easily.
This is crucial for integrating over curved surfaces or volumes which can be tricky in the original variables. By changing variables, we can convert these complicated areas into simpler rectangular regions, making the integration process easier.
When you find a determinant with trigonometric functions, it might look daunting. However, it simply means the transformation concerns rotational or spherical coordinates commonly used in physics and engineering. Using trigonometry in determinants helps to rotate or stretch the space where the function exists, ensuring an accurate change of variables in the integration process.

matrix

A matrix is a way to organize numbers, variables, or expressions into rows and columns. Think of it as a grid or a table. In this exercise, the matrix is a 3x3 layout, meaning it has three rows and three columns.
The elements of the matrix can be functions, like trigonometric expressions, which make calculations seem complex but are part of the puzzle in transformations. In linear algebra, matrices hold a wealth of applications from solving systems of linear equations to geometric transformations.

• Each element in the matrix can influence the determinant calculation significantly, as it represents a transformation rule.
• To solve with matrices, understanding the arrangement allows us to expand and calculate determinants correctly.

This matrix showcases a combination of trigonometric and scalar functions, indicating transformations or rotations in three-dimensional space, crucial for calculating integrals in new coordinate systems.

trigonometry

Trigonometry is part of mathematics that studies relationships between angles and sides of triangles. It uses functions like sine, cosine, and tangent to describe rotations and periodicity.
In the given 3x3 determinant, trigonometric functions appear prominently, transforming how determinants operate in more than just traditional algebraic contexts. These functions allow us to work with angles and distances, often necessary in physics or engineering for rotational dynamics and wave motions.
Consider how sine and cosine appear in rotational matrices. They help in defining angles for rotating systems and can represent physical phenomena like oscillations or waves. This makes them essential in mathematical modeling.

• Sine \( \sin \) and cosine \( \cos \) provide the needed rotational transformation for changing the variable space.
• Applying trigonometric identities can simplify determinant calculations and understanding.

Their clever usage in determinants draws a direct line from abstract mathematics to practical, real-world applications like designing machinery or analyzing rotating systems.