A simple differentiation and partial differentiation

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SUMMARY

The discussion centers on the differentiation of a function V, which is treated differently on the left-hand side as a function of time t, while on the right-hand side, it is a function of multiple variables: t, x, y, and z. The confusion arises from the use of the same symbol V for different contexts, leading to a misunderstanding of simple versus partial differentiation. The mathematical expressions provided, u = dx/dt, v = dy/dt, and w = dz/dt, illustrate the relationships between these variables and clarify the differentiation process.

PREREQUISITES
  • Understanding of basic calculus concepts, specifically differentiation.
  • Familiarity with functions of multiple variables.
  • Knowledge of mathematical notation and symbols used in calculus.
  • Ability to interpret mathematical expressions and their implications.
NEXT STEPS
  • Study the principles of simple differentiation versus partial differentiation.
  • Learn about functions of multiple variables and their derivatives.
  • Explore the concept of notation in mathematics to avoid ambiguity.
  • Practice solving problems involving multiple variables and their rates of change.
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Students of calculus, educators teaching differentiation, and anyone seeking to clarify the distinctions between simple and partial differentiation in mathematical contexts.

mech-eng
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Hi, in the above why is the left-hand side simple differentiation, i.e V is only function of t but in the right it is function of t, x, y, and z. It is very strange that one side is different than the other. Would you like to explain it?

Thank you.
 
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mech-eng said:
View attachment 99698

Hi, in the above why is the left-hand side simple differentiation, i.e V is only function of t but in the right it is function of t, x, y, and z. It is very strange that one side is different than the other. Would you like to explain it?

Thank you.
Mathematically it's clearly not right. It might be a useful exercise to figure out what would be a mathematically sound way to write out what is meant.

The key, I believe, is that the same symbol V is used for two different purposes.
 
u=\frac{dx}{dt},\ v=\frac{dy}{dt},\ w=\frac{dz}{dt}. Does this clarify?
 

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