Directional Derivatives of a vector ----gradient of f(P)----

In summary, the conversation discusses the definition of a derivative of a function with respect to a tangent vector, denoted as v_p[f]. A lemma is also mentioned, stating that if the tangent vector is written as (v_1, v_2, v_3)_P, then the derivative can be expressed as a dot product between the tangent vector and the basis frames. The question of how the gradient term comes into play is solved by understanding it as a dot product and applying the chain rule on the real-valued function f. The book "Differential Geometry" by Erwin Kreyszig is recommended as a helpful resource for learning this topic. A helpful illustration of the lemma is also provided.
  • #1
Ishika_96_sparkles
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TL;DR Summary
Confusion about the gradient term in the derivation.
Definition: Let f be a differentiable real-valued function on ##\mathbf{R}^3##, and let ##\mathbf{v}_P## be a tangent vector to it. Then the following number is the derivative of a function w.r.t. the tangent vector

$$\mathbf{v}_p[\mathit{f}]=\frac{d}{dt} \big( \mathit{f}(\mathbf{P}+ t \mathbf{v}) \big)|_{t=0}$$

Then there is this

Lemma: If ##\mathbf{v}_p= (v_1,v_2,v_3)_P## is a tangent vector to ##\mathbf{R}^3##, then

[tex]\mathbf{v}_p[\mathit{f}]= \sum_i v_i \frac{d \mathit{f}}{dx_i}(P)[/tex]

1) How does this gradient term come into the picture? Since, [itex]\mathit{f}(\mathbf{P}+ t \mathbf{v})[/itex], do we write the argument as [itex]\mathbf{x}(t)[/itex] and then apply the chain-rule in the definition?

2) Could this be thought of as a dot-product ##\mathbf{v}_P \,. \, \nabla \mathit{f}|_p ##?
 
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2) is solved. Indeed its a dot product between the tangent vector and the basis frames.

The question 1) is solved too. The intuition was correct that it has to do with [itex] x_i (t)= p_i+ t v_i [/itex]. The curve along this vector is parameterized by t and the coordinates of the space are a function of this parameter i.e. [itex] \{x_1(t),x_2(t),x_3(t)\}[/itex]. Finally, the chain rule is applied on the real-valued function f as

[tex]\sum_{i=1}^3 \frac{df}{dx_i}\frac{dx_i}{dt}=\sum_{i=1}^3\frac{df}{dx_i}\, v_i[/tex]

I was reading from some pdf notes, earlier. Now, I've found the book to learn it properly.
It is the Differential Geometry by Erwin Kreyszig.
 
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  • #3
Maybe you can also try Hubbard ,Vector Calculus,lineare Algebra and differential forms.It is didactically one of the best books on this level.

Here is nice picture to the Lemma:

image.jpg
 
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  • #4
troglodyte said:
Maybe you can also try Hubbard ,Vector Calculus,lineare Algebra and differential forms.It is didactically one of the best books on this level.

Here is nice picture to the Lemma:

View attachment 264020

Simply WOW!

I just saw the preview and fell in love wit the pedagogical style. Thank you so much for such a wonderful suggestion. :biggrin:
 
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  • #5
Ishika_96_sparkles said:
Simply WOW!

I just saw the preview and fell in love wit the pedagogical style. Thank you so much for such a wonderful suggestion. :biggrin:
I am glad that this Suggestion could help you🙂
 
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What is a directional derivative?

A directional derivative is a measure of the rate of change of a function in a specific direction. It tells us how much the function changes when we move along a specific direction from a given point.

What is the gradient of a function?

The gradient of a function is a vector that points in the direction of the steepest increase of the function at a given point. It is calculated by taking the partial derivatives of the function with respect to each variable.

How do you calculate the directional derivative of a vector?

The directional derivative of a vector can be calculated by taking the dot product of the gradient of the function with the unit vector in the desired direction. This can be represented by the formula Duf(P) = ∇f(P) · u, where ∇f(P) is the gradient of the function and u is the unit vector in the desired direction.

What is the significance of directional derivatives in real life?

Directional derivatives have many applications in real life, particularly in fields such as physics, engineering, and economics. They can be used to analyze the rate of change of physical quantities, optimize designs, and predict the behavior of systems.

What is the relationship between directional derivatives and partial derivatives?

Directional derivatives and partial derivatives are closely related. The partial derivatives of a function represent the rate of change of the function in each direction, while the directional derivative represents the rate of change in a specific direction. The directional derivative can be thought of as a combination of the partial derivatives in each direction.

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