Results 1 to 3 of 3

Math Help - first and second order partial derivative easy questions

  1. #1
    Junior Member
    Joined
    Jan 2011
    Posts
    52

    first and second order partial derivative easy questions

    It has been a long time since I've had to do derivatives so I'm rusty.

    show that \frac {\delta u}{\delta t} = k \frac {\delta ^2 u}{\delta x^2} where 'k' is constant.

    u=u(x,t)= exp(-n^2kt)sin(nx)

    I get :
    u_t(x,t)=-n^2k(1) exp(-n^2kt)sin(nx)

    and

    u_x(x,t)=exp(-n^2kt) ncos(nx)

    u_(xx)(x,t)=exp(-n^2kt) (-n^2sin(nx))

    so if I multiply the second order derivative (U_xx) by 'k' then this should be correct?

    I just want to make sure that since I'm doing partial derivatives with respect to x for the last 2 that I can ignore the exp(-n^2kt) because it is all a cosntant. And when I do the derivative of sin(nx) I only need to pull the 'n' out and change it to cos(nx). Also, for the first derivative with respect to 't' I am taking the '1' exponent off of 't' and bringing it forward.

    Last semester was 100% integrals and I feel like I have to relearn the first semester all over. Thanks for the help.

    Also, just notices that my U_xx doesnt work. Anyone able to tell me how to get two sub x's in a row. Figured parathese would work, guess not.
    Last edited by Bowlbase; October 16th 2012 at 07:27 PM. Reason: -n looked like I was subtracting instead of multiplying. ( ) to make clear
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member MaxJasper's Avatar
    Joined
    Aug 2012
    From
    Canada
    Posts
    482
    Thanks
    54

    Lightbulb Re: first and second order partial derivative easy questions

    \frac{\partial ^2u(x,t)}{\partial x^2} = n^2 \left(-e^{-k n^2 t}\right) \sin (n x)

    \frac{\partial u(x,t)}{\partial t} = k n^2 \left(-e^{-k n^2 t}\right) \sin (n x)
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Mar 2010
    Posts
    993
    Thanks
    244

    Re: first and second order partial derivative easy questions

    You use braces: U_{xx} to give U_{xx}. To get the other notation, do \frac{\partial^2U}{\partial{x}^2} for \frac{\partial^2U}{\partial{x}^2}. The braces around x in the denominator are just to separate it from the word "partial". A space works, too: \frac{\partial^2U}{\partial x^2} gives the same thing.

    When you take the partial derivative with respect to x, t is held constant, so exp(-n^2kt) is a constant and you're just differentiating the sine function. The second derivative works the same way. So you go from sine to cosine to negative sine, and two factors of n come out.

    When you take the partial derivative with respect to t, x is held constant, so \sin{nx} is a constant any you're just differentiating the exponential, and a factor of -n^2k comes out. So you have the same thing as U_{xx} with an additional factor of k.

    When I take the derivative of e^{ax}, my thinking is that I bring down the coefficient of whatever I'm differentiating with respect to. Technically, I'm using the chain rule, so I'm multiplying by the derivative of ax.

    - Hollywood
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Second order partial derivative question
    Posted in the Calculus Forum
    Replies: 0
    Last Post: March 18th 2012, 11:12 AM
  2. Replies: 1
    Last Post: April 22nd 2010, 02:11 AM
  3. Partial Derivative Questions
    Posted in the Calculus Forum
    Replies: 2
    Last Post: February 27th 2010, 03:12 PM
  4. Higher Order Partial Derivative
    Posted in the Calculus Forum
    Replies: 1
    Last Post: June 9th 2009, 03:30 AM
  5. A second order partial derivative
    Posted in the Calculus Forum
    Replies: 8
    Last Post: October 19th 2007, 03:52 PM

Search Tags


/mathhelpforum @mathhelpforum