MATH 162: Calculus II Framework for Thurs., Mar
MATH 162: Calculus II Framework for Thurs., Mar. 29–Fri. Mar. 30 The Gradient Vector Today’s Goal: To learn about the gradient vector ∇~ f and its uses, where f is a function of two or three variables. The Gradient Vector Suppose f is a differentiable function of two variables x and y with domain R, an open region of the xy-plane. Suppose also that r(t) = x(t)i + y(t)j, t ∈ I, (where I is some interval) is a differentiable vector function (parametrized curve) with (x(t), y(t)) being a point in R for each t ∈ I. Then by the chain rule, d ∂f dx ∂f dy f(x(t), y(t)) = + dt ∂x dt ∂y dt 0 0 = [fxi + fyj] · [x (t)i + y (t)j] dr = [fxi + fyj] · . (1) dt Definition: For a differentiable function f(x1, . , xn) of n variables, we define the gradient vector of f to be ∂f ∂f ∂f ∇~ f := , ,..., . ∂x1 ∂x2 ∂xn Remarks: • Using this definition, the total derivative df/dt calculated in (1) above may be written as df = ∇~ f · r0. dt In particular, if r(t) = x(t)i + y(t)j + z(t)k, t ∈ (a, b) is a differentiable vector function, and if f is a function of 3 variables which is differentiable at the point (x0, y0, z0), where x0 = x(t0), y0 = y(t0), and z0 = z(t0) for some t0 ∈ (a, b), then df ~ 0 = ∇f(x0, y0, z0) · r (t0). dt t=t0 • If f is a function of 2 variables, then ∇~ f has 2 components.
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