Chapter 12: Q. 25 (page 976)

In Exercises 21–26, find the discriminant of the given function.
$f (θ, ϕ) = \cos θ \sin ϕ$ .

Short Answer

Expert verified

The answer is $\sin^{2} ϕ \cos^{2} θ - \cos^{2} ϕ \sin^{2} θ$ .

Step by step solution

Step 1. Given Information.

The function is $f (θ, ϕ) = \cos θ \sin ϕ$ .

Step 2. Explanation.

The discriminant is calculated by formula,

$d e t (H_{f}) = \frac{\partial^{2} f}{\partial θ^{2}} \cdot \frac{\partial^{2} f}{\partial ϕ^{2}} - {(\frac{\partial^{2} f}{\partial θ \partial ϕ})}^{2}$ .

Calculate $\frac{\partial f}{\partial θ}$ , $\frac{\partial f}{\partial \emptyset}$ , $\frac{\partial^{2} f}{\partial θ^{2}}$ , $\frac{\partial^{2} f}{\partial ϕ^{2}}$ and $\frac{\partial^{2} f}{\partial θ \partial ϕ}$ .

$\frac{\partial f}{\partial θ} = - \sin θ \sin ϕ$ , $\frac{\partial^{2} f}{\partial θ^{2}} = - \cos θ \sin ϕ$ .

localid="1650041459981" $\frac{\partial f}{\partial \emptyset} = \cos ϕ \cos θ$ , $\frac{\partial^{2} f}{\partial ϕ^{2}} = - \sin ϕ \cos θ$ .

$\frac{\partial^{2} f}{\partial θ \partial ϕ} = - \cos ϕ \sin θ$ .

Step 3. Calculation.

Calculate $d e t (H_{f}) = \frac{\partial^{2} f}{\partial θ^{2}} \cdot \frac{\partial^{2} f}{\partial ϕ^{2}} - {(\frac{\partial^{2} f}{\partial θ \partial ϕ})}^{2}$ .

role="math" localid="1650042389735" $d e t (H_{f}) = (- \cos θ \sin ϕ) (- \sin ϕ \cos θ) - {(- \cos ϕ \sin θ)}^{2} = \sin^{2} ϕ \cos^{2} θ - \cos^{2} ϕ \sin^{2} θ$

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In Exercises 21–26, find the discriminant of the given function.
$f (θ, ϕ) = \cos θ \sin ϕ$ .

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

Step 3. Calculation.

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In Exercises 21–26, find the discriminant of the given function.fθ,ϕ=cosθsinϕ.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

Step 3. Calculation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Logic and Functions

Mechanics Maths

Statistics

Calculus

Pure Maths

Study anywhere. Anytime. Across all devices.

In Exercises 21–26, find the discriminant of the given function.
$f (θ, ϕ) = \cos θ \sin ϕ$ .