Discussion Overview
The discussion revolves around the mathematical properties of the gradient operator in the context of electric fields, specifically examining the relationship between the gradient of scalar functions and vector multiplication. Participants explore specific cases involving the function \( \frac{1}{|r|} \) and its gradient, as well as related expressions.
Discussion Character
- Technical explanation
- Mathematical reasoning
- Debate/contested
Main Points Raised
- One participant suggests that taking the negative gradient of \( \frac{1}{|r|} \) results in \( \frac{\hat{\mathbf{r}}}{|r|^2} \), indicating a specific case of the gradient operator.
- Another participant confirms the previous claim regarding the gradient of \( \frac{1}{|r|} \) and poses a question about the gradient of \( \frac{2}{|r|} \), asking if it would yield \( \frac{2\hat{\mathbf{r}}}{|r|^2} \).
- A subsequent post reiterates the question about the gradient of \( \frac{2}{|r|} \) and seeks clarification on the mathematical expression, indicating uncertainty about the notation used.
- Another participant asserts that a multiplicative constant can be factored out of the gradient, implying a general property of the gradient operator.
Areas of Agreement / Disagreement
Participants generally agree on the gradient of \( \frac{1}{|r|} \) but there is uncertainty regarding the gradient of \( \frac{2}{|r|} \), with some participants seeking clarification and others affirming the mathematical properties involved.
Contextual Notes
There are unresolved aspects regarding the notation and specific mathematical steps in the discussion, particularly in relation to the expressions used and the assumptions made about the functions involved.
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with V being the same as psi. Doing the algebra, I would get (Q/4πε0)*(r-hat/|r|2)=-∇(Q/4πε0r)-∂A/∂t. Now in the case that A does not change with time, we would get that taking the negative gradient is equal to multiplying by r-hat/|r|. Is this correct?