Chapter 4: Q4MP (page 238)

Find $\frac{\partial^{2} r}{\partial θ \partial y}$ .

Short Answer

Expert verified

The value of $\frac{\partial^{2} r}{\partial θ \partial y} is \cos θ$

Step by step solution

Given Information

Given that $r^{2} = x^{2} + y^{2}$ and the polar coordinates are $x = r \cos θ, y = r \sin θ$ .

find the value of ∂2r∂θ∂y

The value of r is expressed as,

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

The above equation differentiates with respect to y.

$\frac{\partial r}{\partial y} = \frac{y}{\sqrt{x^{2} + y^{2}}} \frac{\partial r}{\partial y} = \frac{r \sin θ}{\sqrt{r^{2}}} \frac{\partial r}{\partial y} = \sin θ \$

Now,

$\frac{\partial^{2} r}{\partial θ \partial y} = \frac{\partial}{\partial θ} (\frac{\partial r}{\partial y}) \frac{\partial^{2} r}{\partial θ \partial y} = \frac{\partial}{\partial θ} (\sin θ) \frac{\partial^{2} r}{\partial θ \partial y} = \cos θ$

Hence the value of $\frac{\partial^{2} r}{\partial θ \partial y} = \cos θ$ .

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Find $\frac{\partial^{2} r}{\partial θ \partial y}$ .

Short Answer

Step by step solution

Given Information

find the value of ∂2r∂θ∂y

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Find ∂2r∂θ∂y.

Short Answer

Step by step solution

Given Information

find the value of ∂2r∂θ∂y

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Most popular questions from this chapter

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Find $\frac{\partial^{2} r}{\partial θ \partial y}$ .