Partial Derivatives: What are They and Why Are They Used?

In summary, the ability to determine if a function's first and second partial derivatives are continuous is dependent on having a formula for B or assuming that B is the magnetic field based on empirical laws of physics. This is because most naturally occurring phenomena are infinitely differentiable.
  • #1
sparkle123
175
0
Why can you do this?
6f2ca8b6.png

thanks!
 
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  • #2
It can be done if B, as a function of x & t, and its first and second partial derivatives are all continuous.
 
  • #3
Thanks!
How would you know if a function's first and 2nd partial derivatives are continuous?
 
  • #4
sparkle123 said:
Thanks!
How would you know if a function's first and 2nd partial derivatives are continuous?

If you have a formula for B, you can try and evaluate them.

If B is the magnetic field, you can assume so, because our empirical laws of physics say so.
(Most naturally occurring phenomena are infinitely differentiable.)
 
  • #5
Yes, B is the magnetic field! Thanks! :)
 

Related to Partial Derivatives: What are They and Why Are They Used?

1. What are partial derivatives?

Partial derivatives are a type of derivative used in multivariable calculus to measure the rate of change of a function with respect to one variable while holding all other variables constant.

2. Why are partial derivatives used?

Partial derivatives are used to analyze how a function changes in response to changes in its input variables. They are particularly useful in fields such as physics and economics, where multiple variables are often involved.

3. How are partial derivatives calculated?

Partial derivatives are calculated by holding all but one variable constant, taking the derivative of the function with respect to that variable, and then replacing the constant variables with their values.

4. What is the difference between partial derivatives and ordinary derivatives?

Partial derivatives are used for functions with multiple variables, while ordinary derivatives are used for functions with a single variable. Additionally, partial derivatives only consider the rate of change in one variable, whereas ordinary derivatives consider the overall rate of change of the function.

5. In what fields are partial derivatives commonly used?

Partial derivatives are commonly used in fields such as physics, economics, engineering, and statistics. They are also used in machine learning and data analysis to optimize functions with multiple variables.

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