Lesson 4

Partial Differentiation

Topics

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  • Euler theorem

    2 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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    3 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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    5 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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