Directional Derivative for Linear Maps

In summary: You can also use the chain rule to simplify this equation.In summary, the directional derivative of a linear map is the partial derivative of the function with respect to the canonical basis vectors. If you are working with a canonical map, then the directional derivative is equivalent to the matrix equation of the map.
  • #1
asif zaidi
56
0
Hi:

Can someone point how to approach this problem- we had 5 problems on directional derivatives and I solved 4. I understand the concept but in this question I don't know where to begin

Problem Statement
Assume that f:R[tex]^{n}[/tex] -> R[tex]^{m}[/tex] is a linear map, with matrix A with respect to the canonical bases. Show that Df(xo) = f for every xo [tex]\in [/tex] R[tex]^{n}[/tex]


Plz advise - I will probably post follow-up questions to any answers

Thanks

Asif
 
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  • #2
asif zaidi said:
I understand the concept but in this question I don't know where to begin
You should never be at a loss of how to begin a problem -- definitions are almost always a reasonable starting point.
 
  • #3
Specifically, what is the definition of "Df" and what happens when you apply that definition to a linear map?
 
  • #4
I will give what I have done so far...

Definition of a directional derivative is its partial derivatives wrt to all the variables in the given function.

So in this case the question is for f:R[tex]^{n}[/tex]->R[tex]^{m}[/tex] there is an mxn matrix which would look like the following

Df(x0) = (assume this is equation 1)

df1/dx1 df1/dx2... df1/dxn
df2/dx1 df2/dx2... df2/dxn
. ... . ... .
. ... . ... .
dfm/dx1 dfm/dx2...dfm/dxn

Where in above matrix dfm/dx1 is the partial derivative of function wrt x1,x2... I couldn't find symbol for partial derivative

Now, if I use the definition of a linear map then I know that
D([tex]\alpha1[/tex]+[tex]\alpha2[/tex] ) f(x0) = [tex]\alpha1[/tex]Df(x0) + [tex]\alpha2[/tex] Df(x0)

I can also prove by continuity and as t->0 and [tex]\varsigma[/tex]->0 that
Df(x0 + [tex]\varsigma[/tex]p1 + t[tex]\alpha2[/tex]p2) -> Df(x0) ----- equation 2

Now since this is a canonical map the above matrix of Df(x0) in equation 1 reduces to the following
1 0... 0
0 1... 0
. ...
. ...
0 0...1

So in equation2 since D is essentially the above matrix, I can say the following:
Df(x0 + [tex]\varsigma[/tex]p1 + t[tex]\alpha2[/tex]p2) -> f(x0) which is what I think the question wants.

Is this correct?

Thanks

Asif
 
  • #5
Yes.

(in LaTex, [ tex ]\partial[ /tex ] gives [tex]\partial[/tex].)
 

1. What is a directional derivative?

A directional derivative is a measure of how a function changes along a specific direction in a multi-dimensional space. It is a generalization of the ordinary derivative in calculus, which measures the rate of change of a function in a specific direction.

2. How is the directional derivative calculated?

The directional derivative is calculated by taking the dot product of the gradient vector of the function and the unit vector representing the direction in which the derivative is being measured. It can also be calculated using partial derivatives in the direction of the vector.

3. What is the significance of the directional derivative in real-world applications?

The directional derivative is used in various fields such as physics, engineering, and economics to model the rate of change of a function in a specific direction. It can help determine the direction of maximum change or the steepest descent/ascent for a given function.

4. Can the directional derivative be negative?

Yes, the directional derivative can be negative if the function is decreasing in the direction of the vector or if the gradient vector and the direction vector have opposite directions. A negative directional derivative indicates a decrease in the function value with respect to the direction being measured.

5. How does the directional derivative relate to the concept of a slope?

The directional derivative can be thought of as the slope of a tangent line in multi-dimensional space. Just like how the slope of a tangent line represents the rate of change of a function in one dimension, the directional derivative measures the rate of change of a function in a specific direction in multi-dimensional space.

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