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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

Note by Aritra Jana
1 year ago

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@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

- 8 months, 2 weeks ago

Approaches

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

Staff - 8 months, 2 weeks ago

What have you done? What have you tried?

Staff - 1 year ago

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

- 1 year 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?

- 1 year ago

- 1 year ago