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Math Help - multivariable calc, derivatives

  1. #1
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    Exclamation multivariable calc, derivatives

    calculate the jacobian matrix for the given function at the indicated point. write a formula for the total derivative.

    H(r, theta)= (rcostheta, rsintheta) a= (square root of 2, 3pi/2)

    so i changed it to H(x,y) and got the matrix
    1 0
    0 1

    but the answer in the back of the book is
    0 squ.rt 2
    -1 0


    i feel like it has to do with the point a but i don't know where i'm supposed to plug this point in? or how to write the formula for the total derivative?
    sorry this is my first time doing derivatives in this class so i'm a bit confused
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by holly123 View Post
    calculate the jacobian matrix for the given function at the indicated point. write a formula for the total derivative.

    H(r, theta)= (rcostheta, rsintheta) a= (square root of 2, 3pi/2)

    so i changed it to H(x,y) and got the matrix
    1 0
    0 1

    but the answer in the back of the book is
    0 squ.rt 2
    -1 0


    i feel like it has to do with the point a but i don't know where i'm supposed to plug this point in? or how to write the formula for the total derivative?
    sorry this is my first time doing derivatives in this class so i'm a bit confused
    The Jacobian matrix in this case is \mathcal{J}\!\left(r,\theta\right)=\begin{bmatrix}  \frac{\partial}{\partial r}r\cos\theta & \frac{\partial}{\partial \theta}r\cos \theta\\ \frac{\partial}{\partial r}r\sin \theta & \frac{\partial}{\partial \theta}r\sin \theta\end{bmatrix}=\begin{bmatrix}\cos\theta &-r\sin\theta\\ \sin\theta & r\cos\theta\end{bmatrix}

    Now what is \mathcal{J}\!\left(\sqrt{2},\tfrac{3\pi}{2}\right)?
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  3. #3
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    ohh that makes more sense. i got the answer in the back now. but how do i write a formula for the total derivative? is there a general formula for this i can't seem to find it in the book
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  4. #4
    MHF Contributor Calculus26's Avatar
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    By total derivative are you talking about the total differential?


    By total derivative we can also mean the derivative of a function of several variables with respect to one of the input variables.

    Both formulas are in the attachment

    If you are talking about the Total Derivative of a Transformation if I am not mistaken that is the Jacobian matrix.

    I would like some clarification on that myself. My best guess is based on what I know of differentials for single valued functions is detJ * drd(theta) = rdrd(theta) (areal element for polar coords)
    Attached Thumbnails Attached Thumbnails multivariable calc, derivatives-differential.jpg  
    Last edited by Calculus26; September 28th 2009 at 02:46 AM.
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  5. #5
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    thanks, nevermind i was just supposed to put the jacobian matrix into a function. kind of like in calc one, taking the derivative then finding the equation of the tangent line
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