Proving Formula in Vector Differentiation

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SUMMARY

The discussion centers on proving the vector differentiation formula d(A∙B) = A∙dB + dA∙B. The user attempts to validate this by substituting the left side with the general formula of vector differentiation, resulting in an additional term ΔA∙ΔB. The main challenge is demonstrating that ΔA∙ΔB equals zero, which requires identifying relevant identities in vector differentiation and understanding the limits involved in the differentiation process.

PREREQUISITES
  • Understanding of vector calculus and differentiation
  • Familiarity with the dot product of vectors
  • Knowledge of limits in calculus
  • Concept of infinitesimals in mathematical analysis
NEXT STEPS
  • Research vector differentiation identities and their proofs
  • Study the properties of the dot product in vector calculus
  • Learn about limits and their application in differentiation
  • Explore the concept of infinitesimals and their role in calculus
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Students and professionals in mathematics, physics, and engineering who are looking to deepen their understanding of vector differentiation and its applications in various fields.

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1. The problem statement
The formula that must be proven is:
d (A∙B) = A∙dB + dA∙B
du du du


2. The attempt at a solution
When I substitute the left side of the equation to the general formula of vector differentiation, I got the left side of the equation + ΔA∙ΔB
du
Now my only problem is that how can I prove that ΔA∙ΔB = 0 so that it will result to the above equation? Is there any identity in vector differentiation that is ΔA∙ΔB = 0?
 
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First, what are [itex]\Delta A[/itex] and [itex]\Delta B[/itex]? The "general formula of vector differentiation" that I know involves limits. What limits are you taking?
 

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