Which of the following function is a valid potential function ( $ϕ$ )?

Question

Question

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SJVN ET Mechanical 2019 Official Paper

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Answer 1

Concept:

A function $ \phi(x, y) $ is a valid potential function if its mixed second-order partial derivatives are equal:

$ \frac{\partial^2 \phi}{\partial x \partial y} = \frac{\partial^2 \phi}{\partial y \partial x} $

Check for Option:

$ \phi(x, y) = y^3 - 3x^2y $

$ \frac{\partial \phi}{\partial x} = -6xy \Rightarrow \frac{\partial^2 \phi}{\partial x \partial y} = -6x $

$ \frac{\partial \phi}{\partial y} = 3y^2 - 3x^2 \Rightarrow \frac{\partial^2 \phi}{\partial y \partial x} = -6x $

Since the mixed partials are equal, the function is a valid potential function.