Engineering Mathematics

5

Partial Differentiation

Let n be a positive integer and ℜ be the set of real numbers. Then, ℜ_n is the set of all n-tuples (x₁, x₂, …, x_n), x_n ∈ ℜ. Thus,

ℜ is the set of all real numbers called, the real line,

ℜ² = ℜ × ℜ = {(x, y): x, y ∈ ℜ} is a twodimensional Cartesian plane,

ℜ³ = ℜ × ℜ × ℜ = {(x, y, z): x, y, z ∈ ℜ} is a three-dimensional Euclidean space.

Let A be a nonempty subset of ℜⁿ. Then, a function f : A → ℜis called a real-valued function of n variables defined on the set A. Thus, f maps (x₁, x₂, …, x_n), x_i ∈ ℜ into a unique real number f (x₁, x₂, …, x_n).

A function f of n variables x₁, x₂,…, x_n is said to tend to a limit λ as (x₁, x₂,…x_n) → (a_n, a₂,…a_n) if given ε > 0, however small, there exists a real number δ > 0 such that

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