Tarea para el 12 de Septiembre

1.- Calcular el gradiente de la función

a) f (x,y) = 4x² – 3xy + y²

f’ x (x,y) = 8x – 3y + 0 i

f ’y (x,y) = 0 – 3x + 2y j

Vf(x, y) = (8x-3y) i + (-3x + 2y) j

b) f(x,y,z) = x-y ÷ x+z

f’x(x,y,z)

= (x+z)(1) – (x-y)(1) ÷ (x+z)²

= z+y / (x+z)² i

f’y(x,y,z)

= - (x+z) (d/dx y) – (y) (d/dx x+z) ÷ (x+z)²

= - (x+z) (1) – y (0) ÷ (x+z)²

= - x+z ÷ (x+z)²

= - 1 / x+z j

f’z(x,y,z)

= (x+z) (d/dx x – y) – (x – y) (d/dx x+z) ÷ (x+z)²

= x+2 (0) – (x – y) (1) ÷ (x+z)²

= - x+y / (x+z)² k

Vf(x,y,z) = [z+y/(x+z)²] i + [- 1/x+z ] j [- x+y/ (x+z)²] k

2.- Calcular la divergencia y el rotacional de campo vectorial F

a) F(x, y,z) = 6x² i – xy² j

div F= V.F = M + N +R

div F = 12x – x2y

rot = V x F =

i = + (-x2y)(0) – (xy²)(0) = 0

j = - (12x)(0) – (6x²)(0) = 0

k = + (12x)(-xy²) – (6x²) (-x2y) = (-12x²y²) – (-6x³-2y) = -12x²y² + 6x³2y

rot = (0,0, -12x²y² + 6x³2y)

b) F(x,y,z) = senx i + cosy j + z²

div F = V.F = cosx (1) – seny (1) +2z

div F = cosx – seny +2z

rot = V x F =

i = + (-seny)(z²) – (cosy) (2z) = + (-senyz² – cosy2z) = -senyz² – cosy2z

j = - (cosx)(z²) – (senx)(2z)= - (cosxz² – senx2z) = -cosxz² + senx2z

k = + (cosx)(cosy) – (senx)(-seny) = + (cosxcosy + senxseny) = +cosxcosy + senxseny

rot = -senyz² – cosy2z i -cosxz² + senx2z j +cosxcosy + senxseny k

c) F(x,y,z) = e^xseny i – e^xcosx j En el punto (0,0,3)

divF = e^xseny – 0

divF = e^xseny

rot= V x F =

i = + (0)(0) – (-e^xcosx)(0) = 0

j = - (e^xseny)(0) – (e^xseny)(0) = 0

k = + (e^xseny)(-e^xcosx) –(e^xseny)(0) = +(-e^xseny e^xcosx) = -e⁰sen0 e⁰cos0= -(1)(0)(1)(1)

rot = 0 i + 0 j + 0 k