Find where is the unit vector in the direction of X1X Solut

Find , where is the unit vector in the direction of X_1-X.

Solution

first of all we will find Grad f partial derivative and then multiply by direction vector
(i) Grad f = (pi*y*z cos(pixyz)i + pi*x*z cos(pixyz)j + pi*y*x cos(pixyz)k )* (2i + 1j +2k)
   = pi(2y*z cos(pixyz) + x*z cos(pixyz) + 2y*x cos(pixyz) )
(ii) Grad f = [-2x*e^-(x^2+y^2+2z)i +-2y*e^-(x^2+y^2+2z)j +-2*e^-(x^2+y^2+2z)k] * [ i-2k]
   =(-2x+4)*e^-(x^2+y^2+2z)
(iii) Grad f = [(1/(+x+y+z))i +(1/(+x+y+z))j + (1/(+x+y+z))k ]*(2i-2k)
   = 0
(iv) Grad f = 4x^3(i+j+k+l....)*(i+j+k+l...)
   4x^3*n