Functions of Multiple Variables and Partial Differentiation

• In science and engineering, functions often depend on multiple variables
• The ideal gas law, p = ρRT, shows pressure (p) as a function of density (ρ) and temperature (T)
• R represents the gas constant and remains material property, not a variable

Functions of Multiple Variables

• The volume (V) of a right circular cone, V = (1/3)πr²h, depends on the radius (r) and altitude (h)

Partial Differentiation

• It involves taking the derivative with respect to a single variable while keeping other independent variables constant
• The symbol ∂ is used to denote partial derivatives

Function of Two Variables

• If f associates every (x, y) in a region D of the xy-plane with a unique real number z, z = f(x, y) is a function of two variables x and y
• x and y are called independent variables, while z is the dependent variable

Neighborhood of a Point (a, b)

• For any positive number δ, points (x, y) such that a − δ ≤ x ≤ a+δ and b − δ ≤ y ≤ b+δ form a square
• The square is bounded by lines x = a − δ, x = a+δ, y = b – δ, and y = b+δ, and its center is at (a, b)
• This square is called a neighborhood of the point (a, b)

Continuity of a Function of Two Variables

• A real-valued function w = f(x, y) defined in a region D is considered continuous at a point (x₀, y₀) ∈ D if lim(x,y)→(x₀,y₀) f(x, y) = f(x₀, y₀)
• If f is continuous at every point in D, it is continuous in D

Continuity Example

• A function f(x, y) = xy / √(x² + y²) when (x, y) ≠ (0, 0) and 0 when (x, y) = (0, 0) is continuous at the origin
• By converting to polar coordinates (x = r cos θ, y = r sin θ), it can determined that the limit as (x, y) approaches (0, 0) is 0, which equals f(0, 0)

Limit Requirement

• For the limit lim(x,y)→(x₀,y₀) f(x, y) = f(x₀, y₀) to exist, f(x, y) must approach f(x₀, y₀) along every path of (x, y) to (x₀, y₀)

Discontinuity Example

• The function f(x, y) = xy / (x² + y²) when (x, y) ≠ (0, 0) and 0 when (x, y) = (0, 0) is discontinuous at the origin
• As (x, y) approaches (0, 0) along y = x, the limit is 1/2
• As (x, y) approaches (0, 0) along y = -x, the limit is -1/2
• Since the limits differ, the overall limit does not exist, and the function is discontinuous

Partial Derivatives of Two Independent Variables

• If z = f(x, y), keeping y constant (y = b) allows f(x, b) to depend on x alone
• If the derivative of f(x, b) exists at x = a, this is the partial derivative with respect to x at (a, b), denoted as (∂z/∂x)(a,b) or (∂f/∂x)(a,b)
• The partial derivative is defined by the limit (if it exists): ∂f/∂x = lim(∆x→0) [f(a + ∆x, b) − f(a, b)] / ∆x
• The partial derivative with respect to y is defined similarly: ∂f/∂y = lim(∆y→0) [f(a, b + ∆y) – f(a, b)] / ∆y

Function of x and y Points

• The partial derivatives ∂f/∂x and ∂f/∂y depend on the point (a, b)
• At a variable point (x, y), the partial derivatives ∂f/∂x and ∂f/∂y are functions of x and y

Calculating Partial Derivatives

• To find ∂f/∂x, differentiate z = f(x, y) with respect to x, treating y as a constant
• To find ∂f/∂y, differentiate z with respect to y, treating x as a constant

Geometric Interpretation of Partial Derivatives

• The function z = f(x, y) can be visualized as a surface in space
• Setting y = b represents a vertical plane intersecting the surface in a curve
• The partial derivative ∂f/∂x at a point (a, b) gives the slope of the tangent line to the curve
• Specifically, it is equivalent to tan α, where α is the angle relative to where α is the angle as in figure 1
• Similarly, ∂f/∂y at (a, b) is the slope of the tangent to the curve x = a on the surface z = f(x, y)

Higher Order Derivatives

• Partial derivatives of first order from ∂z/∂x and ∂z/∂y
• Differentiating these once more gives partial derivatives of second order

Shorthand Notation

• Partial derivatives can be written as
• ∂z/∂x = p
• ∂z/∂y = q
• ∂²z/∂x² = r
• ∂²z/∂x∂y = s
• ∂²z/∂y∂x = s
• ∂²z/∂y² = t

Higher Order Partial Derivatives

• Third and higher orders can be defined
• Example notation: fxy = ∂²z/∂x∂y
• If the partial derivatives are continuous, the order of differentiation is immaterial
• Implying fxy = fyx, fxxy = fxyx = fyxx, and fxyy = fyyx = fyxy

Order of Partial Differentiation

• Calculating partial derivatives in different orders can vary in difficulty
• When calculating ∂²f/∂y∂x, finding ∂f/∂x first may be complex; finding ∂f/∂y first can simplify the expression

Homogeneous Functions

• A function u = f(x, y) is homogeneous of degree n if it can be expressed as u = xⁿφ(y/x) or u = yⁿψ(x/y)

Homogeneous Functions Expression

• The expression a₀xⁿ + a₁xⁿ⁻¹y + a₂xⁿ⁻²y² + … + aₙyⁿ, where every term is of the nth degree, represents a homogeneous function of degree n
• This can be written as xⁿ[a₀ + a₁(y/x) + a₂(y/x)² + ... + aₙ(y/x)ⁿ]

General Definition

• A function f(x, y, z, t, ...) is homogeneous of degree n if it can be expressed as xⁿφ(y/x, z/x, t/x, .....)

Homogenous Function Examples

• Functions x² tan⁻¹(y/x) and x³ cos(y/x)

Euler's Theorem on Homogeneous Functions

• If u is an homogeneous function of degree n in x and y, then x(∂u/∂x) + y(∂u/∂y) = nu

Corollary to Euler's Theorem

• If u is a homogeneous function of degree n, then x²(∂²u/∂x²) + 2xy(∂²u/∂x∂y) + y²(∂²u/∂y²) = n(n − 1)u

Total Derivative Definition

• If u = f(x, y) where x = φ(t) and y = ψ(t), then the ordinary derivative du/dt, known as the total derivative of u, is given by du/dt = (∂u/∂x)(dx/dt) + (∂u/∂y)(dy/dt)

Implicit Function

• If f(x, y) = c is an implicit function, its total derivative is given by 0 = (∂f/∂x) + (∂f/∂y)(dy/dx)
• The formula for the second differential coefficient of an implicit function is d²y/dx² = -1/q³[q²r – 2pqs + p²t]

Taylor’s Series in 2-Dimensions

• If the function f(x1, x2, ...., xn) has continuous first partial derivatives with respect to each of its are continuously differentiable functions of t, then are continuously differentiable functions of t
• By assumptions of theorem g(t) possesses continuous derivatives of order n in the neighboorhood of origin