Partial Differentiation

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

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

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

3 = 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, …, xn), xi ∈ ℜ into a unique real number f (x1, x2, …, xn).

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


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