Partial differential = the change?

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SUMMARY

The discussion clarifies the relationship between the finite difference notation Δy/Δx and the partial derivative notation ∂y/∂x in calculus. It establishes that Δy/Δx approximates ∂y/∂x when Δy and Δx are sufficiently small, particularly when treating y as a constant while differentiating with respect to x. The conversation also emphasizes that if x and y are independent variables, then ∂y/∂x equals zero. This understanding is essential for grasping the fundamentals of multivariable calculus.

PREREQUISITES
  • Understanding of basic calculus concepts, including derivatives and limits.
  • Familiarity with multivariable functions, specifically functions of independent variables.
  • Knowledge of finite difference approximations in calculus.
  • Concept of partial derivatives and their notation.
NEXT STEPS
  • Study the concept of limits in calculus to better understand the transition from finite differences to derivatives.
  • Learn about multivariable calculus, focusing on partial derivatives and their applications.
  • Explore the implications of independent variables in multivariable functions.
  • Investigate the use of Taylor series for approximating functions in multiple dimensions.
USEFUL FOR

Students of calculus, mathematics educators, and anyone seeking to deepen their understanding of multivariable calculus and the relationship between finite differences and partial derivatives.

rsaad
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partial differential = the change??

Homework Statement



How is
Δy/Δx = [itex]\partial y[/itex] / [itex]\partial x[/itex] ?

I just don't know the logic behind this.
 
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That is only approximately or asymptotically true. If Δy and Δx are small Δy/Δx will be near ∂y / ∂x.
 


rsaad said:

Homework Statement



How is
Δy/Δx = [itex]\partial y[/itex] / [itex]\partial x[/itex] ?

I just don't know the logic behind this.
Do you understand how [itex]\Delta y/\Delta yx\approx dy/dx\ ?[/itex]

Usually, x & y are independent variables -- and often there are additional independent variables.

If f is a function of independent variables, x and y, then we write f(x,y).

[itex]\partial f/\partial x\ \[/itex] is essentially [itex]\ \ df/dx\ \[/itex] if we treat y as being held at some fixed value.

Then [itex]\ \ \Delta f/\Delta x \approx \partial f/\partial x\ \[/itex] keeping y fixed at some value.



By the Way: If x & y are independent variables, then [itex]\ \ \partial y/\partial x=0 \ .[/itex]
 

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