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The inner product of two column vectors \mathbf a and \mathbf b, \mathbf a \cdot \mathbf b, written in matrix form is \mathbf a^T * \mathbf b.
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Definition of Partial Derivative: Correct Form?
- Context: Graduate
- Thread starter Cyrus
- Start date
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- Definition
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Discussion Overview
The discussion revolves around the correct definition of the partial derivative, specifically the representation of the gradient as either a row vector or a column vector. Participants explore the implications of these definitions in mathematical proofs and their usage in various texts.
Discussion Character
- Debate/contested
- Mathematical reasoning
Main Points Raised
- Some participants note that one definition represents the gradient as a row vector while the other represents it as a column vector.
- One participant suggests that the context of the notation is crucial, as it resembles the gradient operator.
- Another participant argues that the notation df/d\vec{x} is a non-standard way to express the gradient, which should be represented as a row vector.
- Some participants express confusion over the use of the bar in the notation and its implications for differentiating scalars with respect to vectors.
- There is a suggestion that the correct definition may depend on the conventions used in different textbooks or courses.
- One participant emphasizes that the gradient should be represented as a row vector because it transforms differently than displacement vectors, which are typically column vectors.
- Another participant expresses frustration over the inconsistency in definitions across different sources, highlighting the need for clarity in notation.
- Some participants discuss the relationship between the gradient and the differential of a function, suggesting that they are not completely equivalent.
- There is a contention regarding whether spatial coordinates should be treated as column or row vectors, with differing opinions on the implications of each choice.
Areas of Agreement / Disagreement
Participants do not reach a consensus on which definition is correct, with multiple competing views remaining on the representation of the gradient and the implications of different notational conventions.
Contextual Notes
Participants highlight that the choice of representation may depend on the specific context or convention used in various mathematical texts, leading to confusion and disagreement.
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