Arg such that for any nonzero complex number z, Arg(z) is the uniquely determined argument of z lying in the interval (−π, π].
Examples: Arg(−1) = π, Arg(−1 + i) = 3π/4, and Arg(−i) = − π/2
Given z = x + iy is nonzero, where x and y are real and y ≥ 0, we can think of Arg(z) as the angle between the vector ⟨x, y⟩ and the direction of the positive x axis (specified by say, the vector ⟨1, 0⟩). Similarly, given z = x + iy is nonzero, where x and y are real and y < 0,we can think of Arg(z) as minus the angle between the vector ⟨x, y⟩ and the direction of the positive x axis. Thus Arg(z) can be interpreted as giving the direction of z.
A smooth curve C is the complex plane is the graph of a function f : [a, b] → C such that a < b and f′ is continuous and nonzero at each t in [a,b]
f′ is continuous and nonzero on [a,b].
The assumption that f′ is nonzero, means that the point f(t), a ≤ t ≤ b moves along the curve C, with increasing t, from the initial point f(a) to the final (or terminal) point f(b). Thus, we can associate with a smooth curve a direction of traversal (from the initial point to the final point). The curve C is closed if the initial point and final points are the same f(a) = f(b). The curve C is simple provided that for a < t1 < t2 < b, f(t1) /= f(t2), meaning that the the curve does not cross itself.
∇φ(x_1, x_2, ..., x_n) = ⟨φ_x1 (x1, x2, ..., xn), φ_x2 (x_1, x_2, ..., x_n), ..., φ_xn (x_1, x_2, ..., x_n)⟩ Let f = f(x_1, x_2, ..., x_n) be a real-valued function of n real variables, let a ̄ = (a_1, a_2, ..., a_n) be an interior point of its domain and let u ̄ = (u_1, u_2, ..., u_n) be a unit vector in R^n. Then directional derivative of f at a ̄ in the direction of u ̄, denoted (D_u(f))(ā), is given by...