Partial Differentiation of Composite Function |u = f(x/y, y/z, z/x), u = f(x−y, y−z, z−x)
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Partial Differentiation of Composite Function |u = f(x/y, y/z, z/x), u = f(x−y, y−z, z−x)
88 324 просмотра · 3 г. назад
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88 324 просмотра · 3 г. назад
📘 Calculus & Multivariable Calculus | Partial Differentiation Problem Solutions
📝 *Problem 1:*
If
u = f(x/y, y/z, z/x),
prove that
x(∂u/∂x) + y(∂u/∂y) + z(∂u/∂z) = 0.
🕘 *Problem 2 (at 9:00):*
If
u = f(x − y, y − z, z − x),
prove that
(∂u/∂x) + (∂u/∂y) + (∂u/∂z) = 0.
🎯 *Concepts Used:*
✔️ Partial Differentiation
✔️ Chain Rule for Multivariable Functions
✔️ Euler’s Theorem Type Relations
✔️ Simplification using Symmetry of Variables
✨ These problems beautifully illustrate how functions involving variable ratios or differences exhibit differential identities that simplify elegantly using multivariable calculus concepts.
✅ Very useful for Engineering Mathematics, Partial Differentiation, Multivariable Calculus, and VTU exam preparation.
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