\(\large 1.\) 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

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TopNewest@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

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Approaches

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What have you done? What have you tried?

Can you show your working?

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

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

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@Otto Bretscher

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

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