Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I Calculate partial derivatives and mixed partial derivatives

  1. Oct 5, 2016 #1

    dyn

    User Avatar

    Hi. I know how to calculate partial derivatives and mixed partial derivatives such as ∂2f/∂x∂y but I've now become confused about something. If I have a function of 3 variables eg. f(x,y,z) and I calculate ∂x then I am differentiating wrt x while holding y and z constant. Does that mean ∂x then becomes a function of x only ie does ∂x f(x,y,z) = φ( x ) ? If it does then ∂y and ∂z will always be zero but I know this is not the case. I'm confused !
     
  2. jcsd
  3. Oct 5, 2016 #2

    fresh_42

    Staff: Mentor

    No. It's still a function of three variables.
    Differentiating along a single variable simply means to consider the change in values along this coordinate.

    You might want to play around a little with Wolfram, e.g. http://www.wolframalpha.com/input/?i=f(x,y)+=+xy^3++4+x^2
    Imagine the partial derivation ##\partial_x## as a tangent in ##x-##direction. It still varies with ##y## and ##z##.
     
  4. Oct 5, 2016 #3

    dyn

    User Avatar

    Thanks. So it is incorrect to write ∂x f(x, y ,z) = g ( x ) ?
     
  5. Oct 5, 2016 #4

    FactChecker

    User Avatar
    Science Advisor
    Gold Member

    Unless the variables y and z disappear as a result of differentiating wrt x, they are still there. They might disappear, but don't count on it.
     
  6. Oct 5, 2016 #5

    fresh_42

    Staff: Mentor

    Yes.

    You could of course evaluate the derivative at some point ##p=(x_0,y_0,z_0)## and get
    $$\frac{\partial}{\partial_x}\bigg{|}_p f(x,y,z) = g(x_0,y_0,z_0)$$
    Or if you want to examine the ##x-##component, you could consider ##g(x,y_0,z_0)## and get a function in one variable, because you fixed ##y=y_0## and ##z=z_0##. But this is another issue.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted