Partial Derivatives in Differential Calculus: Need Help

In summary, partial derivatives are derivatives of a function with respect to one of its variables while holding all other variables constant. They are essential in differential calculus as they allow us to analyze the behavior of multivariable functions. To calculate a partial derivative, you first need to identify the variable and treat all others as constants. The main difference between partial derivatives and total derivatives is that partial derivatives only consider change in one direction, while total derivatives take into account change in all directions. Partial derivatives are commonly used in fields such as physics, engineering, economics, and statistics to analyze complex systems with multiple variables.
  • #1
jem05
56
0
im trying to get a function R^2 --> R
whose partial derivatives of all orders exist in a neighborhood of (0,0) but it, itself, is not continuous at the origin.
thing is, i think I am thinking of the question wrong because, I am getting examples and checking if partial der. exist but how can i ensure that all the orders do so,...?
any ideas? thx
 
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  • #2
Let f(x,y) = 0 at the origin,
= (cos t , sin t) otherwise ( where x = Rcost ,y =Rsint).
 
  • #3
isnt that the same as f(r,t) = cost , with 0 if r = 0?
 

1. What are partial derivatives?

Partial derivatives are derivatives of a function with respect to one of its variables while holding all other variables constant. They are used to measure the rate of change of a function in a specific direction.

2. Why do we need to use partial derivatives in differential calculus?

Partial derivatives are essential in differential calculus because they allow us to analyze the behavior of multivariable functions. They help us understand how a function changes in relation to each of its variables, which is crucial in many scientific and mathematical applications.

3. How do you calculate partial derivatives?

To calculate a partial derivative, you first need to identify the variable that you are taking the derivative with respect to. Then, you treat all other variables as constants and use the standard rules of differentiation to find the derivative. This process may need to be repeated for each variable in the function.

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

The main difference between partial derivatives and total derivatives is that partial derivatives only consider the change of a function in one direction, while total derivatives take into account the change in all directions. Partial derivatives are used for multivariable functions, while total derivatives are used for single-variable functions.

5. In what fields of science are partial derivatives commonly used?

Partial derivatives are used in many fields of science, including physics, engineering, economics, and statistics. They are particularly useful in analyzing the behavior of complex systems with multiple variables, such as fluid dynamics, thermodynamics, and optimization problems.

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