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 ntuples (x_{1}, x_{2}, …, x_{n}), x_{n} ∈ ℜ. Thus,
ℜ is the set of all real numbers called, the real line,
ℜ^{2} = ℜ × ℜ = {(x, y): x, y ∈ ℜ} is a twodimensional Cartesian plane,
ℜ^{3} = ℜ × ℜ × ℜ = {(x, y, z): x, y, z ∈ ℜ} is a threedimensional Euclidean space.
Let A be a nonempty subset of ℜ^{n}. Then, a function f : A → ℜis called a realvalued function of n variables defined on the set A. Thus, f maps (x_{1}, x_{2}, …, x_{n}), x_{i} ∈ ℜ into a unique real number f (x_{1}, x_{2}, …, x_{n}).
A function f of n variables x_{1}, x_{2},…, x_{n} is said to tend to a limit λ as (x_{1}, x_{2},…x_{n}) → (a_{n}, a_{2},…a_{n}) if given ε > 0, however small, there exists a real number δ > 0 such that

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