Taylor expansion, of gradient of a function, in multiple dimensions

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Discussion Overview

The discussion revolves around the Taylor expansion of a multidimensional function, specifically focusing on the gradient and the integral form of the expansion. Participants are exploring the mathematical formulation and derivation of these concepts, as well as seeking clarification on specific terms and their meanings.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant presents a form of the Taylor expansion for a multidimensional function and expresses understanding of the initial part but seeks clarification on the integral term.
  • Another participant questions the nature of the multiplication in the gradient term, wondering if it is a dot product between two vectors.
  • A third participant corrects a notation issue regarding the gradient vector, suggesting the inclusion of a transpose.
  • Participants share resources, including a lecture slide and a Wikipedia article, that may provide further context on the topic.

Areas of Agreement / Disagreement

There is no consensus on the understanding of the integral term in the Taylor expansion, as participants express confusion and seek clarification. Multiple viewpoints on the nature of the multiplication in the gradient term are also present.

Contextual Notes

Participants have not resolved the specific mathematical details regarding the integral form of the Taylor expansion or the nature of the multiplication in the gradient term. The discussion includes references to external resources that may contain relevant information.

Whenry
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Hello all,

I understand that the taylor expansion for a multidimensional function can be written as

f(\overline{X} + \overline{P}) = f(\overline{X}) + \nabla f(\overline{X}+t\overline{P})(\overline{P})

where t is on (0,1).

Although I haven't seen that form before, it makes sense.

But I don't understand the integral in the following the Taylor expansion,

\nabla f(\overline{X} + \overline{P}) = \nabla f(\overline{X}) + \int^{1}_{0} \nabla^{2} f(\overline{X}+t\overline{P})(\overline{P})dt

Could someone help me understand the derivation?

Thank you,

Will
 
Last edited:
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Whenry said:
Hello all,

I understand that the taylor expansion for a multidimensional function can be written as

f(\overline{X} + \overline{P}) = f(\overline{X}) + \nabla f(\overline{X}+t\overline{P})(\overline{P})

where t is on (0,1).

I don't understand it. What kind of multiplication is going on in the last term? It appears to be two vectors multiplied together. Is it a dot product?


But I don't understand the integral in the following the Taylor expansion,

\nabla f(\overline{X} + \overline{P}) = \nabla f(\overline{X}) + \int^{1}_{0} \nabla^{2} f(\overline{X}+t\overline{P})(\overline{P})dt

I don't either, but this is an interesting formula and I would like to know where you saw it. Is this from a subject like fluid dynamics? Can you give a link to a page?
 

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