Need Help Solving these problems of Partial Differentiation

$\large 1.$ GIVEN: $\large F(v^2-x^2,v^2-y^2,v^2-z^2)=0$

where $\large v$ is a function of $x,y,z$

$\textbf{SHOW THAT:}$ $\large \frac{1}{x}\frac{\partial v}{\partial x}+\frac{1}{y}\frac{\partial v}{\partial y}+\frac{1}{z}\frac{\partial v}{\partial z}=\frac{1}{v}$

$\large 2.$ GIVEN: $V=f(X,Y,Z)$ is a homogeneous function in $x,y,z$ of degree $\boxed{n}$

where $X=\dfrac{\partial V}{\partial x}, Y=\dfrac{\partial V}{\partial y}, Z=\dfrac{\partial V}{\partial z}$

$\textbf{SHOW THAT:}$ $\large X\frac{\partial V}{\partial X}+Y\frac{\partial V}{\partial Y}+Z\frac{\partial V}{\partial Z}=\frac{n}{n-1}V$

Now for the second one, i understand that the Euler's theroem is necessary but i can't proceed in a definite direction from there

#Calculus

Easy Math Editor

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Comments

@Calvin Lin @Chew-Seong Cheong @Agnishom Chattopadhyay @Parth Lohomi

Aritra Jana - 4 years, 7 months ago

@Aritra Jana sorry but I don't know much about partial derivatives.

Parth Lohomi - 4 years, 7 months ago

oh sorry, my bad. although , could you help me by tagging some who you think have a certain degree of knowledge in this field?

Aritra Jana - 4 years, 7 months ago

@Otto Bretscher

Parth Lohomi - 4 years, 7 months ago

What have you done? What have you tried?

Can you show your working?

Calvin Lin Staff - 4 years, 7 months ago

@Aritra Jana @Calvin Lin i am opening brilliant after 3 years. i think i have 2 solutions to problem 1 but i am not being able to post a picture.plz help

Arkajyoti Maity - 4 years, 3 months ago

Approaches

If you email it to me, I can upload it for you.
If you upload it to an image site like Imgur, you can then use the markedown syntax of ![](URL) to display images.

Calvin Lin Staff - 4 years, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$