Differentials of multivariable functions

Differentials of multivariable functions

Lessons

Notes:

Differentials of 2 Variable Functions
Suppose there is a 2-variable function z=f(x,y)z=f(x,y). Then we say that the differential dzdz is:

dz=fxdx+fydydz = f_xdx + f_ydy


Differentials of 3 Variable Functions
Of course, differentials can be extended to 3-variable functions as well. Suppose there is a 3-variable function w=g(x,y,z)w=g(x,y, z). Then we say that the differential dwdw is:

dw=gxdx+gydy+gzdzdw = g_x dx + g_y dy + g_z dz

  • Introduction
    Differentials of Multivariable Functions Overview:
    a)
    A Review of Differentials
    • dy,dxdy, dx are differentials
    • dy=f(x)dxdy=f'(x)dx

    b)
    Differentials of 2 Variable Functions
    • Differential dzdz
    • dz=fxdx+fydydz=f_xdx+f_ydy
    • An example

    c)
    Differentials of 3 Variable Functions
    • Differential dw dw
    • dw=gxdx+gydy+gzdzdw=g_xdx+g_ydy+g_zdz
    • An example


  • 1.
    Finding Differentials of 2 Variable Functions
    Compute the differential for the following function:

    f(x,y)=ex3+y3sinx2f(x,y) = e^{x^3+y^3} \sin \frac{x}{2}


  • 2.
    Compute the differential for the following function:

    z=ln(x2y32)z = ln (\frac{x^2 y^3 }{2} )


  • 3.
    Finding Differentials of 3 Variable Functions
    Compute the differentials for the following function:

    g(x,y,z)=tan[ln(xy2z3)]g(x,y,z) = \tan [ \ln (xy^2 z^3) ]


  • 4.
    Compute the differentials for the following function:

    g=exyxy2z2g = \frac{e^{xy}}{xy^2z^2}