Chapter 12: Q 64. (page 965)

Use at least two methods to prove that $\frac{d r}{d t} = 0$ when $r = \sqrt{x^{2} + y^{2}}$ $x = α \cos t, and y = α \sin t$ if $α$ is constant.

Short Answer

Expert verified

Above relation is proved using chain rule.

$\frac{d r}{d t} = \frac{\partial r}{\partial x} \frac{d x}{d t} + \frac{\partial r}{\partial y} \frac{d y}{d t}$

Step by step solution

Given Information

It is given that

$r = \sqrt{x^{2} + y^{2}}, x = α \cos t and y = α \sin t$

Applying Chain Rule

Using chain rule

$\frac{d r}{d t} = \frac{\partial r}{\partial x} \frac{d x}{d t} + \frac{\partial r}{\partial y} \frac{d y}{d t}$

Solving for $\frac{\partial r}{\partial x}$

$\frac{\partial r}{\partial x} = \frac{\partial}{\partial x} (\sqrt{x^{2} + y^{2}})$

$= \frac{1}{2} {(x^{2} + y^{2})}^{- \frac{1}{2}} \frac{\partial}{\partial x} (x^{2} + y^{2})$

$= \frac{1}{2 \sqrt{x^{2} + y^{2}}} (2 x + 0)$

$= \frac{x}{\sqrt{x^{2} + y^{2}}}$

Finding $\frac{\partial r}{\partial y}$

$\frac{\partial r}{\partial y} = \frac{\partial}{\partial y} (\sqrt{x^{2} + y^{2}})$

$= \frac{1}{2} {(x^{2} + y^{2})}^{- \frac{1}{2}} \frac{\partial}{\partial y} (x^{2} + y^{2})$

$= \frac{1}{2 \sqrt{x^{2} + y^{2}}} (0 + 2 y)$

$= \frac{y}{\sqrt{x^{2} + y^{2}}}$

Differentiating w.r.t t

$\frac{d x}{d t} = \frac{d}{d t} α \cos t = - α \sin t$

Also

$\frac{d y}{d t} = \frac{d}{d t} α \sin t = α \cos t$

Solving for $\frac{d z}{d t} = \frac{\partial z}{\partial x} \frac{d x}{d t} + \frac{\partial z}{\partial y} \frac{d y}{d t}$

$= (\frac{x}{\sqrt{x^{2} + y^{2}}}) (- α \sin t) + (\frac{y}{\sqrt{x^{2} + y^{2}}}) (α \cos t)$

$= \frac{α}{\sqrt{x^{2} + y^{2}}} (- x \sin t + y \cos t)$

$= \frac{α}{\sqrt{(α \cos t)^{2} + (α \sin t)^{2}}} (- α \cos t \sin t + α \sin t \cos t)$

$= \frac{α}{\sqrt{α^{2} (\cos^{2} t + \sin^{2} t)}} \cdot 0 = 0$

Solve using direct derivative

Use $x = α \cos t and y = α \sin t in r = \sqrt{x^{2} + y^{2}}$

Hence,

$r = \sqrt{x^{2} + y^{2}}$

$= \sqrt{(α \cos t)^{2} + (α \sin t)^{2}}$

$= \sqrt{α^{2}}$

Differentiating $r = α$ wrt $t$

$\frac{d r}{d t} = \frac{d}{d t} α = 0$

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Use at least two methods to prove that $\frac{d r}{d t} = 0$ when $r = \sqrt{x^{2} + y^{2}}$ $x = α \cos t, and y = α \sin t$ if $α$ is constant.

Short Answer

Step by step solution

Given Information

Applying Chain Rule

Differentiating w.r.t t

Solve using direct derivative

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Use at least two methods to prove that drdt=0when r=x2+y2x=αcost,andy=αsintif α is constant.

Short Answer

Step by step solution

Given Information

Applying Chain Rule

Differentiating w.r.t t

Solve using direct derivative

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Decision Maths

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Geometry

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Use at least two methods to prove that $\frac{d r}{d t} = 0$ when $r = \sqrt{x^{2} + y^{2}}$ $x = α \cos t, and y = α \sin t$ if $α$ is constant.