Cálculo de Funções de Várias Variáveis - Definições e Teoremas Essenciais

Definition A function f of two variables is a rule that assigns to each ordered pair of real numbers x y in a set D a unique real number denoted by fx y The set D is the domain of f and its range is the set of values that f takes on that is fx y x y D Definition If f is a function of two variables with domain D then the graph of f is the set of all points x y z in R3 such that z fx y and x y is in D Definition The level curves of a function f of two variables are the curves with equations fx y k where k is a constant in the range of f 4 If f is a function of two variables its partial derivatives are the functions fx and fy defined by fxx y lim h0 fxh y fx y h fyx y lim h0 fx yh fx y h Notations for Partial Derivatives If z fx y we write fxx y fx f x x fx y z x f1 D1 f Dxf fyx y fy f y y fx y z y f2 D2 f Dyf 2 Definition The directional derivative of f at x0 y0 in the direction of a unit vector u a b is Dufx0 y0 lim h0 fx0 ha y0 hb fx0 y0 h if this limit exists Dufx y fx y u 15 Theorem Suppose f is a differentiable function of two or three variables The maximum value of the directional derivative Dufx is fx and it occurs when u has the same direction as the gradient vector fx 19 Fxx0 y0 z0x x0 Fyx0 y0 z0y y0 Fzx0 y0 z0z z0 0 The normal line to S at P is the line passing through P and perpendicular to the tangent plane The direction of the normal line is therefore given by the gradient vector Fx0 y0 z0 and so by Equation 1253 its symmetric equations are x x0 Fxx0 y0 z0 y y0 Fyx0 y0 z0 z z0 Fzx0 y0 z0 For a differentiable function of two variables z fx y we define the differentials dx and dy to be independent variables that is they can be given any values Then the differential dz also called the total differential is defined by dz fxx y dx fyx y dy zx dx zy dy EXAMPLE a If z fx y x2 3xy y2 find the differential dz b If x changes from 2 to 205 and y changes from 3 to 296 compare the values of Δz and dz 15 Theorem Suppose f is a differentiable function of two or three variables The maximum value of the directional derivative Dufx is fx and it occurs when u has the same direction as the gradient vector fx 2 The Chain Rule Case 1 Suppose that z fx y is a differentiable function of x and y where x gt and y ht are both differentiable functions of t Then z is a differentiable function of t and dzdt fx dxdt fy dydt 3 The Chain Rule Case 2 Suppose that z fx y is a differentiable function of x and y where x gs t and y hs t are differentiable functions of s and t Then zs zx xs zy ys and zt zx xt zy yt