- #1

Math Amateur

Gold Member

MHB

- 3,998

- 48

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

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

Last edited: