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Higher Order PD Example-1 Find the second partial derivatives of f(x, y) = x3 + x2y3 – 2y2 Mixed Partial Derivatives (Clairaut’s Theorem ) Clairaut’s Theorem sometimes called Schwarz's theorem • Partial derivatives of order 3 or higher can also be defined. For instance, and using Clairaut’s Theorem we can show that fxyy = fyxy = fyyx if these functions are continuous. Example-2 Calculate fxxyz if f(x, y, z) = sin(3x + yz). Partial Differential Equations • Partial derivatives occur in partial differential equations that express certain physical laws. • For the partial differential equation 2 instance, 2 u u 2 0 2 x y is called Laplace’s equation. • Solutions of this equation are called harmonic functions and play a role in problems of heat conduction, fluid flow, and electric potential. Example-3 Show that the function u(x, y) = ex sin y is a solution of Laplace’s equation: Wave equation u 2 u a 2 2 t x 2 2 describes the motion of a waveform, which could be an ocean wave, sound wave, light wave, or wave traveling along a string. A pulse traveling through a string with fixed endpoints as modeled by the wave equation Spherical waves coming from a point source A solution to the 2D wave equation Example-4 Verify that the function u(x, t) = sin(x – at) satisfies the wave equation. Class work Is there a case when mixed PD are not equal? -Yes, where the continuity condition dropped Example: