Lesson 4

Partial Differentiation

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    1 If $u=x^2-y^2 \; , \; v=2xy \; and \; z=f(u,v) $ prove the following

    $ (\dfrac{\partial z}{\partial x})^2 + (\dfrac{\partial z}{\partial y})^2 \;=\; 4\sqrt{u^2+v^2} \bigg[ (\dfrac{\partial z}{\partial u})^2 + (\dfrac{\partial z}{\partial v})^2 \bigg] \ \; \ \; \ \; \ $

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    3 If $ u \;=\; f \bigg( \dfrac{x-y}{xy} , \dfrac{z-x}{xz} \bigg) $ prove the following

    prove that

    $ x^2 \dfrac{\partial u}{\partial x} + y^2 \dfrac{\partial u}{\partial y} + z^2 \dfrac{\partial u}{\partial z} \;=\; 0 $

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    5 If $ \dfrac{x^2}{1+u}+\dfrac{y^2}{2+u}+\dfrac{z^2}{3+u}=1 $ then prove the following

    Prove : $ \bigg(\dfrac{\partial u}{\partial x} \bigg)^2 + \bigg(\dfrac{\partial u}{\partial y} \bigg)^2 + \bigg(\dfrac{\partial u}{\partial z} \bigg)^2 \;=\; 2 \bigg( x\dfrac{\partial u}{\partial x} + y\dfrac{\partial u}{\partial y} + z \dfrac{\partial u}{\partial z} \bigg) \\ \; \\ $

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    14 State and prove Euler theorem for a homogeneous function in two variables and hence find the value of following :

    $x^2 \dfrac{\partial^2 u}{\partial x^2}+ 2xy\dfrac{\partial^2 u}{\partial x \partial y}+ y^2\dfrac{\partial^2 u}{\partial y^2}+ x\dfrac{\partial u}{\partial x}+ y \dfrac{\partial u}{\partial y} \ \; \ \; \ For \; u= e^{x+y} \; + \; log(x^3+y^3-x^2y-xy^2) \ \; \ $

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    16 Euler Theorem : If $ x+y=2e^{\theta} cos\phi \;, \; \; x - y \;=\; 2ie^{\theta} sin\phi \; \;$and u is a function of $x$ and $y$

    then prove that $ \dfrac{\partial^2u}{\partial \theta^2} + \dfrac{\partial^2u}{\partial \phi^2} \;=\; 4xy \dfrac{\partial^2 u}{\partial x \partial y} $

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    20 State and prove Euler's theorem for three variables and hence find the following

    $x \dfrac{\partial u}{\partial x} + y \dfrac{\partial u}{\partial y} + z \dfrac{\partial u}{\partial z} $ where $ u \;=\; \dfrac{x^3y^3z^3}{x^3+y^3+z^3} $

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