Chapter 12: TF. 4 (page 946)

$A chain rule for functions of two variables : Let f (x, y) = x^{2} y^{3}, x = s \cos t, and y = s \sin t . Substitute the expressions for x and y into f (x, y) .$

Short Answer

Expert verified

$f (x, y) = s^{5} \cos^{2} t \sin^{3} t$

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. Substitute the expressions for x and y into f(x, y)

$f (x, y) = x^{2} y^{3}, x = s \cos t, and y = s \sin t \Rightarrow f (x, y) = {(s \cos t)}^{2} \times {(s \sin t)}^{3} \Rightarrow f (x, y) = (s^{2} \cos^{2} t) \times (s^{3} \sin^{3} t) \Rightarrow f (x, y) = s^{5} \cos^{2} t \sin^{3} t$

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$A chain rule for functions of two variables : Let f (x, y) = x^{2} y^{3}, x = s \cos t, and y = s \sin t . Substitute the expressions for x and y into f (x, y) .$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Substitute the expressions for x and y into f(x, y)

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Achainruleforfunctionsoftwovariables:Letf(x,y)=x2y3,x=scost,andy=ssint.Substitutetheexpressionsforxandyintof(x,y).

Short Answer

Step by step solution

Step 1. Given information

Step 2. Substitute the expressions for x and y into f(x, y)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Discrete Mathematics

Calculus

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Geometry

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

$A chain rule for functions of two variables : Let f (x, y) = x^{2} y^{3}, x = s \cos t, and y = s \sin t . Substitute the expressions for x and y into f (x, y) .$