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Differentiating expressions involving multivariable vector valued functions

  1. Oct 16, 2008 #1
    Not really a homework problem... just a general question (this seemed like the place to put it...). Say I have three functions:

    [tex]f,g,h:\mathbb{R}^2\rightarrow\mathbb{R}^3[/tex]

    and an expression along the lines of:

    [tex]\left\langle f(u_1,u_2),g(u_1,u_2)\right\rangle h(u_1,u_2)[/tex]

    What differentiation rules allow me to compute

    [tex]\frac{\partial}{\partial \vec{u}}(\left\langle f(u_1,u_2),g(u_1,u_2)\right\rangle h(u_1,u_2))[/tex]

    My problem is that I'm unclear on how to order/transpose the Jacobians and vectors such that all of the multiplications make sense. It's possible to order them as follows:

    [tex]\left\langle f(\vec{u}),g(\vec{u})\right\rangle \frac{\partial h}{\partial \vec{u}} + h(\vec{u})\cdot f(\vec{u})^\mathrm{T}\cdot \frac{\partial g}{\partial \vec{u}} + h(\vec{u})\cdot g(\vec{u})^\mathrm{T}\cdot \frac{\partial f}{\partial \vec{u}}[/tex]

    Everything works out there, and the result is a 3x2 matrix. However I'm not clear on what rules allow me to actually arrive at this result (other than moving things around until it all fits).

    If anyone knows a good (preferably online) resource or book that discusses these sorts of expressions, that would be very helpful as well.

    Thanks!
     
  2. jcsd
  3. Oct 16, 2008 #2

    HallsofIvy

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    No rules. You have to go back to the definition. Your function is from R2 to R3 so the derivative, at a given point in R2 is the linear transformation from R2 to R3 that most closely approximates f- if f is differentiable so that is unique.

    In a particular coordinate system, so that [itex]F(u_1,u_2)= \left< f(u_1,u_2), g(u_1,u_2), h(u_1,u_2)\right>[/itex] the derivative, at [itex](u_1, u_2)[/itex] can be represented by the 2 by 3 matrix
    [tex]\left[\begin{array}{cc}\frac{\partial f}{\partial u_1} & \frac{\partial f}{\partial u_2} \\ \frac{\partial g}{\partial u_1} & \frac{\partial g}{\partial u_2} \\ \frac{\partial h}{\partial u_1} & \frac{\partial h}{\partial u_2}\end{array}\right][/tex]
    Strictly speaking, the derivative is the linear transformation represented by that matrix.

    Again, that is at a particular point in R2- at a particular (u1, u2). If you want the derivative FUNCTION, you will have to think of it as a function from R2 to the set of matrices from R2 to R3 which can, itself, be represented as a function from R2 to R6.
     
  4. Oct 16, 2008 #3
    Sorry, I must not have been clear. The three functions [tex]f[/tex], [tex]g[/tex], and [tex]h[/tex] are each from [tex]\mathbb{R}^2\rightarrow\mathbb{R}^3[/tex]. That is, they each have coordinate functions [tex]f_i[/tex], [tex]g_i[/tex], and [tex]h_i[/tex] ([tex]f[/tex], [tex]g[/tex], and [tex]h[/tex] are not the coordinate functions of some [tex]F:\mathbb{R}^2\rightarrow\mathbb{R}^3[/tex]). The use of the angle bracket notation was to signify an inner product, not different components of a vector. Thus [tex]\left\langle f(u_1,u_2),g(u_1,u_2)\right\rangle[/tex] is a scalar, and [tex]\left\langle f(u_1,u_2),g(u_1,u_2)\right\rangle h(u_1,u_2)[/tex] is a column vector, and I'm wondering how to compute

    [tex]\frac{\partial}{\partial \vec{u}}(\left\langle f(u_1,u_2),g(u_1,u_2)\right\rangle h(u_1,u_2))[/tex]

    and what differentiation rules apply.

    Thanks!
     
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