Kronecker Delta and Gradient Operator

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SUMMARY

The discussion centers on the relationship between the gradient operator and the Kronecker delta in the context of scalar functions. Specifically, it clarifies that the expression ## \frac {\partial x'_k} {\partial x'_i} ## is equivalent to the Kronecker delta, denoted as ## \delta_{ik} ##. This equivalence holds true because when indices i and k are equal, the derivative equals 1, while it equals 0 when they are not equal. The discussion emphasizes the importance of understanding this relationship in Cartesian coordinates.

PREREQUISITES
  • Understanding of gradient operators in vector calculus
  • Familiarity with the Kronecker delta notation
  • Basic knowledge of scalar functions
  • Concepts of partial derivatives
NEXT STEPS
  • Study the properties of the Kronecker delta in mathematical contexts
  • Learn about gradient operators in vector calculus
  • Explore Cartesian coordinates and their implications in calculus
  • Investigate the application of partial derivatives in physics and engineering
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Students and professionals in mathematics, physics, and engineering who are looking to deepen their understanding of vector calculus and the application of the gradient operator.

maxhersch
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I am looking at an explanation of the gradient operator acting on a scalar function ## \phi ##. This is what is written:
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In the steps 1.112 and 1.113 it is written that ## \frac {\partial x'_k} {\partial x'_i} ## is equivalent to the Kronecker delta. It makes sense to me that if i=k, then the expression is equal to 1 but why would it be 0 if they are not equal? Perhaps I'm not looking at it the right way but any explanation would be appreciated.

Thanks.
 
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Just think about Cartesian coordinates. What is ## \frac{\partial x}{\partial y} ## equal to?
 

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