IGNOU PHE 4 2018 - Determine the direction in which the scalar field φ ( x, y) = xy 2

Question

Determine the direction in which the scalar field φ ( x, y) = xy ² + x³ y

increases the

fastest at the point (1, 2).

25 Jan 2018

Answer :

Word Count : 307

We are asked to determine the direction in which the scalar field $$ \phi(x, y) = xy^2 + x^3 y $$ increases fastest at the point $(1, 2)$. Let's solve this manually, step by step. --- ### Step 1: Recall the gradient formula The direction in which a scalar field increases fastest is given by the gradient vector: $$ \nabla \phi(x, y) = \left( \frac{\partial \phi}{\partial x}, \frac{\partial \phi}{\partial y} \right) $$ --- ### Step 2: Compute partial derivatives 1. Partial derivative with respect to $x$: $$ \frac{\partial \phi}{\partial x} = \frac{\partial}{\partial

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