Differential Calculus Overview

What is the gradient ∇𝜙 of the temperature distribution 𝜙 at the point (1, -2, -1)?

What are the divergence and curl of the vector field 𝑽 = (𝒙𝒚𝒛)𝒊 + (𝟑𝒙𝟐 𝒚)𝒋 + (𝒙𝒛𝟐 − 𝒚𝟐 𝒛)𝒌 at the point (2, -1, 1)?

What is the relation between 𝑢 = 𝑥 + 𝑦 and 𝑣 = 𝑥/𝑦?

How do you estimate the value of the function 𝑓(𝑥, 𝑦) = 𝑦𝑥 when 𝑥 = 1.03 and 𝑦 = 1.02 using Taylor series?

If 𝑓(𝑐𝑥 − 𝑎𝑧, 𝑐𝑦 − 𝑏𝑧) = 0, what is the relation between 𝑎, 𝑏, and 𝑐?

Show that if 𝑢 = 𝑥³ + 𝑦³ + 𝑧³ + 3𝑥𝑦𝑧, then ∂𝑢/∂𝑥 + ∂𝑢/∂𝑦 + ∂𝑢/∂𝑧 = 3𝑢.

Verify that 𝑢 = tan^(-1)((𝑥 − 𝑦)) is a homogeneous function.

Evaluate ∂𝑢/∂𝑥 + 𝑦 ∂𝑢/∂𝑦 for 𝑢 = tan^(-1)((𝑥 − 𝑦)).

Evaluate 𝑥² + 𝑦² + 𝑧 ∂²𝑢/∂𝑥∂𝑦 + ∂²𝑢/∂𝑦∂𝑧 for 𝑢 = tan^(-1)((𝑥 − 𝑦)).

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.

This quiz covers key concepts in differential calculus, focusing on temperature distribution, gradient evaluations, vector calculus properties, and variable dependencies. Additionally, it explores Taylor series expansions in relation to growth rates and scaling factors. Test your understanding through practical problem-solving questions.