Unit 3: Partial Differentiation and Curve Geometry

Partial Differentiation

Definition

When a function z = f(x, y) depends on more than one independent variable (e.g., x and y), we can find its derivative with respect to one variable while treating all other variables as constants. This is called a **partial derivative**.

• Partial derivative with respect to x:
  ∂f/∂x = fₓ(x, y) = lim (h→0) [f(x + h, y) - f(x, y)] / h
  (Treat 'y' as a constant and differentiate 'x' normally).
• Partial derivative with respect to y:
  ∂f/∂y = fᵧ(x, y) = lim (k→0) [f(x, y + k) - f(x, y)] / k
  (Treat 'x' as a constant and differentiate 'y' normally).

Example:

If f(x, y) = 3x²y + 5y³ + sin(x)
∂f/∂x = 6xy + 0 + cos(x) = 6xy + cos(x) (y is constant)
∂f/∂y = 3x² + 15y² + 0 = 3x² + 15y² (x is constant)

Mixed Partial Derivatives

We can take partial derivatives of the partial derivatives. These are called second-order partial derivatives.

• ∂²f/∂x² = fₓₓ = ∂/∂x (∂f/∂x)
• ∂²f/∂y² = fᵧᵧ = ∂/∂y (∂f/∂y)
• ∂²f/∂y∂x = fₓᵧ = ∂/∂y (∂f/∂x) (Differentiate w.r.t. x first, *then* w.r.t. y)
• ∂²f/∂x∂y = fᵧₓ = ∂/∂x (∂f/∂y) (Differentiate w.r.t. y first, *then* w.r.t. x)

Clairaut's Theorem (Symmetry of Mixed Partials): If fₓᵧ and fᵧₓ are both continuous, then they are equal: fₓᵧ = fᵧₓ. The order of differentiation does not matter.

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Euler's Theorem on Homogeneous Functions

Homogeneous Functions

A function f(x, y) is called **homogeneous of degree n** if replacing x with tx and y with ty results in the original function multiplied by tⁿ.

Definition: f(tx, ty) = tⁿ f(x, y)

Example: Let f(x, y) = x³ + 2x²y - y³
f(tx, ty) = (tx)³ + 2(tx)²(ty) - (ty)³
= t³x³ + 2(t²x²)(ty) - t³y³
= t³x³ + 2t³x²y - t³y³
= t³ (x³ + 2x²y - y³) = t³ f(x, y)
Therefore, the function is homogeneous of degree n = 3.

A homogeneous function can always be written in the form f(x, y) = xⁿ F(y/x) or f(x, y) = yⁿ G(x/y).

Euler's Theorem (Statement)

This theorem provides a beautiful relationship between a homogeneous function and its partial derivatives. The syllabus specifies the two-variable case.

Statement: If z = f(x, y) is a homogeneous function of degree n, then:
x(∂f/∂x) + y(∂f/∂y) = n·f

Example: For f(x, y) = x³ + 2x²y - y³ (degree n=3)
∂f/∂x = 3x² + 4xy
∂f/∂y = 2x² - 3y²
x(∂f/∂x) + y(∂f/∂y) = x(3x² + 4xy) + y(2x² - 3y²)
= 3x³ + 4x²y + 2x²y - 3y³
= 3x³ + 6x²y - 3y³
= 3 (x³ + 2x²y - y³) = 3·f
This verifies the theorem, as n = 3.

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Tangents and Normals (Cartesian)

For a curve y = f(x) at a point P(x₁, y₁), the slope of the tangent line is given by the derivative m = dy/dx evaluated at that point.

Equation of Tangent

Using the point-slope form Y - y₁ = m(X - x₁):

Equation of Tangent: Y - y₁ = (dy/dx)·(X - x₁)

Equation of Normal

The normal line is perpendicular to the tangent. Its slope is the negative reciprocal of the tangent's slope, m_normal = -1/m = -1/(dy/dx).

Equation of Normal: Y - y₁ = [-1 / (dy/dx)]·(X - x₁)

Cartesian Subtangent and Subnormal

These are lengths measured along the x-axis.

• Length of Tangent (PT): The segment of the tangent line from the point P to the x-axis. |y₁ * sqrt(1 + (dx/dy)²)|
• Length of Normal (PN): The segment of the normal line from the point P to the x-axis. |y₁ * sqrt(1 + (dy/dx)²)|
• Length of Subtangent (TM): The projection of the tangent segment PT onto the x-axis.

  Length of Subtangent = |y₁ / (dy/dx)| = |y₁ · (dx/dy)|

• Length of Subnormal (MN): The projection of the normal segment PN onto the x-axis.

  Length of Subnormal = |y₁ · (dy/dx)|

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Tangents and Normals (Polar)

For a polar curve r = f(θ), the geometry is different. We are often interested in the angle ϕ (phi) between the radius vector r and the tangent line.

Angle between Radius Vector and Tangent

The angle ϕ is given by the formula:

tan(ϕ) = r / (dr/dθ)

Polar Subtangent and Subnormal

These are lengths on a line drawn *perpendicular* to the radius vector r through the pole (origin).

• Length of Polar Subtangent:

  = |r² / (dr/dθ)|

• Length of Polar Subnormal:

  = |dr/dθ|

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Unit 3: Exam Quick Tips