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Difference of total derivative and partial derivative

  1. Aug 16, 2014 #1
    many books only tell the operation of total derivative and partial derivative,

    so i now confuse the application of these two.

    when doing problem, when should i use total derivative and when should i use partial derivative.
    such a difference is detrimental when doing Physics problem, so i help someone can help me explain the meaning of these two derivatives.
     
  2. jcsd
  3. Aug 21, 2014 #2
    It depends upon the problem. Please provide an example so that I can explain it better. However refer below for a little help:-

    Q.1) Find the rate of change of ideal gas pressure with respect to volume?
    Soln:
    P=nRT/V
    We require rate of change of pressure only with respect to volume, hence we're gonna ignore the other parameters like temp., moles, and treat them as constants. Then comes partial differentiation.
    (∂P/∂V)=(-nRT)/(V^2)

    Q. 2) Calculate the differential change in ideal gas pressure?
    Soln:
    Now total differential is required.
    dP=(nR/V)dT + (RT/V)dn - (nRT/V^2)dV
     
  4. Aug 21, 2014 #3
    Let [itex]f[/itex] be function of three independent variables: [itex]f=f(x,y,z)[/itex]. Partial derivative with respect to one of them is just rate of change of [itex]f[/itex] with respect to that variable.
    As an example, take our function to be potential energy which depends only on a position of a particle: [itex]V=V(x,y,z)[/itex]. If you wanted to know by how much potential energy of your particle changes if you move it just a tiny little bit along the x axis, you would compute [itex]\frac{\partial V}{\partial x} \Delta x[/itex]. Since we decided that V only depends on position, we also have [itex]\frac{\partial V}{\partial t}=0[/itex], because if you changed [itex]t[/itex] a little bit and !held other variables fixed!, nothing would change. That's what we mean when we say that [itex]V[/itex] doesn't depend on time explicitly.
    Now, if we are solving problem in mechanics, let's say, the coordinates [itex]x,y,z[/itex] will in fact be functions of [itex]t[/itex], and finding them will often be your task. Now you can, in a way think of [itex]V[/itex] as implicit function of t. What does it mean? Since coordinates [itex]x,y,z[/itex] are functions of time, you can "plug them into" [itex]V[/itex] function, and get that [itex]V(t)=V(x(t),y(t),z(t))[/itex]. Now as time goes by in mechanical system, the particle moves and the coordinates change. If we now treat [itex]V[/itex] as a function of single variable [itex]t[/itex] in a way I presented, we can compute ordinary derivative known from calculus 1. It is radically different object though - we are not assuming that other variables are held fixed now! That is what some people would call "total derivative" with respect to time.
     
  5. Aug 21, 2014 #4

    HallsofIvy

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    Are you sure the phrase is "total derivative" rather than "total differential"? I think the term "total differential is more common than "total derivative" although I have seen the latter used occasionally (with a meaning different from "total derivative").

    If f(x, y, z) is a function of the three variables, x, y, and z, then the partial derivatives are, of course, [itex]\frac{\partial f}{\partial x}[/itex], [itex]\frac{\partial f}{\partial y}[/itex], and [itex]\frac{\partial f}{\partial z}[/itex]. If, in addition, x, y, and z are themselves all functions of some other variable, t, we could replace each of x, y, and z with its expression as a function of t, reducing f to a function of the single variable t, which then has derivative [itex]\frac{df}{dt}[/itex].

    By the "chain rule" for several variables we have
    [tex]\frac{df}{dt}= \frac{\partial f}{\partial x}\frac{dx}{dt}+ \frac{\partial f}{\partial y}\frac{dy}{dt}+ \frac{\partial f}{\partial z}\frac{dz}{dt}[/tex]

    That is what is called the "total derivative" though, as I said, the "total differential", which would be
    [tex]df= \frac{\partial f}{\partial x}dx+ \frac{\partial f}{\partial y}dy+ \frac{\partial f}{\partial z}dz[/tex]
    is more often used. Notice that there is no dependence on an additional variable, t, so this is much more general.
     
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