Talk:Directional derivative

message: Bold text  —Preceding unsigned comment added by 220.225.127.88 (talk) 07:32, 17 April 2009 (UTC)
thread_title:  differential derivative of a point 

message: I can't make sense of 
: \frac{\partial f}{\partial \mathbf{v}}\cdot\mathbf{u} = Df(\mathbf{v})
I suppose there has something gone wrong --Trigamma (talk) 21:57, 28 May 2010 (UTC)
replies: text: The partial derivative-like notation on the left means the gradient of the scalar-valued function f, which happens to be the total derivative of a scalar-valued function with respect to all its variables. If f were vector-valued, it would be instead the Jacobian, which is the total derivative of one vector with respect to the other. The other notations in that equation make it clear that this is what is meant.--Jasper Deng (talk) 08:33, 23 November 2013 (UTC)
thread_title:  Normal derivative 

message: Minor changes:
#add a couple more sources, 
#introduce better numbering scheme (indented and non-bracketed), 
#made the main properties stand out more,
#made vector notation consistent all the way through (it was a mix of bold + overarrow, typographically bold is slightly easier)
F = q(E+v×B) ⇄ ∑ici 13:23, 13 April 2012 (UTC)
thread_title:  slight clean up 

message: As may be seen from the threads above, there seems to be a lot of confusion about the definition of the directional derivative. It may be noted that the citations given are very weak.
The only sensible definition (i.e. one that is mathematically useful and elegant) does not involve any reference to normalization. My impression is that the most notable definition (i.e. generally used definition in authoritative texts) is this one, though I do not have many of these texts to hand. There are many reasons for this claim, to the extent that I would argue for the relegation of the use of the normalized vector to a footnote. A major argument for this is that in the context of functions over a set of coordinates and related contexts, the norm of a vector is usually not defined but the directional derivative remains an extremely useful concept. Even where an (indefinite) metric tensor is defined as in the physics of general relativity, the definition runs into serious difficulties with no apparent benefits.
My guess is that the weaker definition arose as the tail wagging the dog: a term was needed for the very useful construct that generalizes the partial derivative. An obvious term would be the "directional derivative", even if this may be a partial misnomer, which then was interpreted literally.
I would appreciate authoritative texts being cited by those who have access to them. — Quondum 07:15, 30 October 2012 (UTC)
replies: text: I have an (elementary) textbook defining it using unit vectors, but it only considers steady vector fields and does not go beyond ordinary 3-dimensional space.--Jasper Deng (talk) 08:34, 23 November 2013 (UTC)
thread_title:  Verifiability of definition 

message: It is a small error but I think it should be corrected anyway. In the first plot (the one with the contour plot), the gradient has an index 'u' in its name, but a 'u' in index means that we talk about directional derivative (that is not a vector by the way). I don't know how to replot it but I would do it myself if I knew.
Thanks
--Jotwo (talk) 18:16, 4 April 2014 (UTC)
thread_title:  Error in the first plot 

message: I think some of the quantity types are wrong. For example, the directional derivative of a vector field in the direction of any vector should again be a vector field, not a tensor. Since the directional derivative is the limit of a sequence of quotients of the original function's values by scalars, I think the directional derivative must match the original function's type.--Jasper Deng (talk) 17:48, 29 November 2015 (UTC)
thread_title:  Quantities in the "continuum mechanics" section