Can I pull a time derivative outside of a curl?

In summary, the individual is questioning whether it is correct to pull out the time derivative when taking the curl of both sides of the equation ∇ x E = -∂B/∂t, and the expert confirms that it is indeed correct due to the property of interchangeability of partial derivatives.
  • #1
L_landau
27
0

Homework Statement


For the equation ∇ x E = -∂B/∂t I took the curl of both sides to get

∇ x (∇ x E) = ∇ x -∂B/∂t

I feel like it'd be very wrong to pull out the time derivative. Am I correct?
 
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  • #2
L_landau said:

Homework Statement


For the equation ∇ x E = -∂B/∂t I took the curl of both sides to get

∇ x (∇ x E) = ∇ x -∂B/∂t

I feel like it'd be very wrong to pull out the time derivative. Am I correct?
No, you can pull out the time derivative.
 
  • #3
L_landau said:

Homework Statement


For the equation ∇ x E = -∂B/∂t I took the curl of both sides to get

∇ x (∇ x E) = ∇ x -∂B/∂t

I feel like it'd be very wrong to pull out the time derivative. Am I correct?

Yes, this is just the property of interchangeability of partial derivatives: ##\partial / \partial t \; \partial / \partial \psi f(\psi,t) = \partial /\partial \psi \; \partial / \partial t f(\psi,t)##.
 

What is a time derivative?

A time derivative is a mathematical operation that measures the rate of change of a quantity with respect to time. It is often represented by the symbol "d/dt".

What is a curl?

A curl is a mathematical operation that measures the rotation or circulation of a vector field. It is represented by the symbol "curl" or "∇ x".

Can a time derivative be pulled outside of a curl?

Yes, a time derivative can be pulled outside of a curl in certain situations. This is known as the "Leibniz integral rule" and it states that if the function being differentiated is continuous and the derivative exists, then the derivative can be pulled outside of the integral.

What happens when a time derivative is pulled outside of a curl?

When a time derivative is pulled outside of a curl, it essentially means that the time derivative is being applied to each component of the vector field separately. This can simplify calculations and make it easier to solve equations involving the curl.

Are there any limitations to pulling a time derivative outside of a curl?

Yes, there are limitations to pulling a time derivative outside of a curl. This rule only applies in certain situations and cannot be applied to all types of vector fields. Additionally, the function being differentiated must be continuous and the derivative must exist at that point.

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