The Directional Derivative .... in Scalar Fields and Vector Fields ....

[Math Amateur]

I need some guidance regarding the directional derivative ...

Two books I am reading introduce the directional derivative somewhat differently ... these books are as follows:

Theodore Shifrin: Multivariable Mathematics

and

Susan Jane Colley: Vector Calculus (Second Edition)Colley introduces the directional derivative in the context of scalar fields (real-valued functions of vector variables) and defines the directional derivative as follows:View attachment 7472Colley never raises the concept of a directional derivative for vector fields (vector-valued functions of a vector variable) ... ...Shifrin on the other hand introduces the concept of a directional derivative in the context of a vector field (where the case of a scalar field is a special case where the codomain has dimension \(\displaystyle m = 1\) ...) ... as follows:
View attachment 7473How do we reconcile these differences ... and how is it best to think about the directional derivative ...?

What is the more usual approach ... ..?
Help will be much appreciated ... ...

Peter

 

[Math Amateur]

I need some guidance regarding the directional derivative ...

Two books I am reading introduce the directional derivative somewhat differently ... these books are as follows:

Theodore Shifrin: Multivariable Mathematics

and

Susan Jane Colley: Vector Calculus (Second Edition)Colley introduces the directional derivative in the context of scalar fields (real-values functions of vector variables) and defines the directional derivative as follows:Colley never raises the concept of a directional derivative for vector fields (vector-valued functions of a vector variable) ... ...Shifrin on the other hand introduces the concept of a directional derivative in the context of a vector field (where the case of a scalar field is a special case where the codomain has dimension \(\displaystyle m = 1\) ...) ... as follows:
How do we reconcile these differences ... and how is it best to think about the directional derivative ...?

What is the more usual approach ... ..?
Help will be much appreciated ... ...

Peter

I have located an approach that uses both concepts of "directional derivative" - that is directional derivative for scalar fields and the notion of directional derivative for vector fields ( although the author concerned (Tom Apostol) does not call it that ...) ...The approach I am referring to in this post is from the book ...

Tom M. Apostol: "Calculus: Volume II" (Second Edition)Apostol in Sections 8.6 (pages 252-254) discusses and defines the derivative of a scalar field at a point \(\displaystyle \underline{a}\) and in a direction of the vector \(\displaystyle \underline{y}\) ... ... as

\(\displaystyle f'( \underline{a} ; \underline{y}) = \lim_{ h \rightarrow 0} \frac{ f( \underline{a} + h \underline{y} ) - f ( \underline{a} ) }{h} \)

when the limit exists ... ...

... ... as follows:
View attachment 7474
View attachment 7475
According to Apostol, in the special case where \(\displaystyle \underline{y}\) is a unit vector, \(\displaystyle f'( \underline{a} ; \underline{y} )\) is called the directional derivative ... ... as follows:
View attachment 7476

Apostol then continues in Section 8.18 to define the derivative of a vector-valued function of a vector variable (see below) by an expression that looks to me like it is identical to what Shifrin called a directional derivative (see below) ... Is that right?

***EDIT *** I now note that the limit must exist ... not for a particular \(\displaystyle \underline{y}\) but for every \(\displaystyle \underline{y}\) ... so this may make it different from a directional derivative ...

... ... Apostol, note, does not call it as a "directional derivative" ... but just refers to it as the derivative of a vector field ... as follows:

\(\displaystyle \underline{f'}( \underline{a} ; \underline{y}) = \lim_{ h \rightarrow 0} \frac{ \underline{f}( \underline{a} + h \underline{y} ) - \underline{f}( \underline{a} ) }{h}\)and then goes on to show that \(\displaystyle \underline{f'}( \underline{a} ; \underline{y}) = \left( f'_1 ( \underline{a} ; \underline{y}) , \ ... \ ... \ , f'_m ( \underline{a} ; \underline{y}) \right) \)

where (I think), the \(\displaystyle f'_1 ( \underline{a} ; \underline{y})\) are directional derivatives ... is that right?

and then Apostol shows that \(\displaystyle \underline{f'}( \underline{a} ; \underline{y})\) is equal to the total derivative ... as follows:https://www.physicsforums.com/attachments/7477
https://www.physicsforums.com/attachments/7478
Am I correct in identifying \(\displaystyle \underline{f'}( \underline{a} ; \underline{y})\) and the \(\displaystyle f'_i ( \underline{a} ; \underline{y})\) as "directional derivatives" ... ... ?Peter

 

[HallsofIvy]

You've put so much here, it is hard to tell exactly what you are asking. In your first post, at least, the second "directional derivative", of a vector valued function, is just the obvious generalization of the first, of a scalar valued function. Typically, the first thing one learns in Calculus is to take the derivative of a real valued function over the real numbers, then the derivative of a vector valued function over the real numbers as a vector whose components are the derivatives of the (real valued) component functions. Similarly here. The first definition assigns the gradient vector of a real valued function of several variables. The second assigns, as the derivative of a vector valued function of several variables, the linear operator function (which may be represented as a matrix having the gradient vector of the real valued components of the vector valued function as columns).