Chapter 12: TF. 6 (page 946)

$A chain rule for functions of two variables : Let f (x, y) = x^{2} y^{3}, x = s \cos t, and y = s \sin t . Make a conjecture about the relationship between \frac{\partial f}{\partial s}, \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial x}{\partial s}, and \frac{\partial y}{\partial s} .$

Short Answer

Expert verified

Relationship between given partial derivatives is:

$\frac{\partial f}{\partial s} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial s}$

Step by step solution

Step 1. Given information

$f (x, y) = x^{2} y^{3}, x = s \cos t, and y = s \sin t .$

Step 2. Finding relationship between ∂f∂s, ∂f∂x, ∂f∂y, ∂x∂s , and ∂y∂s

Relationship between given partial derivatives is:

$\frac{\partial f}{\partial s} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial s} L H S = \frac{\partial f}{\partial s} = 5 s^{4} \cos^{2} t \sin^{3} t R H S = \frac{\partial f}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial s} = (2 x y^{3}) (\cos t) + (3 x^{2} y^{2}) (\sin t) = 2 (s \cos t) {(s \sin t)}^{3} (\cos t) + 3 {(s \cos t)}^{2} {(s \sin t)}^{2} (\sin t) = 2 s^{4} \cos^{2} t \sin^{3} t + 3 s^{4} \cos^{2} t \sin^{3} t = 5 s^{4} \cos^{2} t \sin^{3} t$

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Short Answer

Step by step solution

Step 1. Given information

Step 2. Finding relationship between ∂f∂s, ∂f∂x, ∂f∂y, ∂x∂s , and ∂y∂s

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Achainruleforfunctionsoftwovariables:Letf(x,y)=x2y3,x=scost,andy=ssint.Makeaconjectureabouttherelationshipbetween∂f∂s,∂f∂x,∂f∂y,∂x∂s,and∂y∂s.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Finding relationship between ∂f∂s, ∂f∂x, ∂f∂y, ∂x∂s , and ∂y∂s

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Theoretical and Mathematical Physics

Applied Mathematics

Mechanics Maths

Logic and Functions

Decision Maths

Study anywhere. Anytime. Across all devices.