Differential Calculus Overview

Differential Calculus Overview

• Analyze temperature distribution within a composite material using the equation ( \phi = 3x^2y - y^3z^2 ).
• Gradient ( \nabla \phi ) indicates the direction and rate of fastest increase of the temperature function.

Evaluating Gradient

• Calculate ( \nabla \phi ) at the point (1, -2, -1) for the temperature distribution equation.

Vector Calculus

• Given the vector ( \vec{V} = (xyz) \hat{i} + (3x^2y) \hat{j} + (xz^2 - y^2z) \hat{k} ).
• Evaluate both divergence and curl of ( \vec{V} ) at the point (2, -1, 1) to analyze vector field properties.

Dependency of Variables

• Establish the relationship between ( u = x + y ) and ( v = \frac{x - y}{x} ).
• Demonstrate that ( u ) and ( v ) are dependent via their functional representation.

Taylor Series Expansion

• Model function ( f(x, y) = yx ) to relate rate of growth ( x ) and scaling factor ( y ).
• Use Taylor series expansion up to second-degree terms around the point (1, 1) for approximation when ( x = 1.03 ) and ( y = 1.02 ).

Homogeneous Functions

• Given ( f(cx - az, cy - bz) = 0 ), prove ( a + b = c ) using the chain rule in multivariable calculus.

Partial Derivation Equality

• Show ( \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} + \frac{\partial u}{\partial z} = 3u \cdot \frac{\partial u}{\partial x} \cdot \frac{\partial u}{\partial y} \cdot \frac{\partial u}{\partial z} ) where ( u = x^3 + y^3 + z^3 + 3xyz ).

Verification of Homogeneity

• For ( u = \tan^{-1} \left( \frac{x - y}{2} \right) ), confirm ( u ) is a homogeneous function.
• Evaluate ( \frac{\partial u}{\partial x} + y \cdot \frac{\partial u}{\partial y} ) and ( 2 \frac{\partial^2 u}{\partial x^2} + 2 \frac{\partial^2 u}{\partial y^2} + z \cdot \frac{\partial u}{\partial z}^2 ) for properties of partial derivatives.