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Questions and Answers
What is the gradient ∇𝜙 of the temperature distribution 𝜙 at the point (1, -2, -1)?
What is the gradient ∇𝜙 of the temperature distribution 𝜙 at the point (1, -2, -1)?
∇𝜙 = (6, -12, 4)
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 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)?
Divergence = 6, Curl = (0, 0, 0)
What is the relation between 𝑢 = 𝑥 + 𝑦 and 𝑣 = 𝑥/𝑦?
What is the relation between 𝑢 = 𝑥 + 𝑦 and 𝑣 = 𝑥/𝑦?
𝑣 = 𝑢/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?
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?
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.
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.
Verify that 𝑢 = tan^(-1)((𝑥 − 𝑦)) is a homogeneous function.
Evaluate ∂𝑢/∂𝑥 + 𝑦 ∂𝑢/∂𝑦 for 𝑢 = tan^(-1)((𝑥 − 𝑦)).
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)).
Evaluate 𝑥² + 𝑦² + 𝑧 ∂²𝑢/∂𝑥∂𝑦 + ∂²𝑢/∂𝑦∂𝑧 for 𝑢 = tan^(-1)((𝑥 − 𝑦)).
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Study Notes
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.
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