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Mike2
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I'm wondering if physics ever uses a differential equation of the form of a curl of a gradient of a scalar function. Or is this too trivial?
Thanks.
Thanks.
Originally posted by chroot
The curl of the gradient of a scalar function is always zero. So, yeah, it's pretty trivial.
- Warren
I think I see what you're asking. How about electrostatics? The electric field can be represented by a scalar potential field. Therefore, the curl of the gradient of this field is zero -- the electric field is curl-free.Originally posted by Mike2
well yes, but then don't they use the fact that the Laplacian is zero?
Originally posted by Mike2
well yes, but then don't they use the fact that the Laplacian is zero?
Originally posted by Mike2
well yes, but then don't they use the fact that the Laplacian is zero?
Originally posted by Ambitwistor
Yes, but the Laplacian of an arbitrary function isn't automatically zero, so only certain functions (the harmonic ones) satisfy the condition that their Laplacian is zero. Every function satisfies the condition that the curl of its gradient equals zero, so that equation is not too useful on its own.
Originally posted by lethe
physicists use both facts.
In other words, you could never find a "unique" function that satisfies the conditions. For ALL functions satisfy the conditions, right?
Originally posted by Mike2
Yes, but the question is HOW do they use the fact that the curl of the gradient of a scalar. Yes they use it in Stoke's theorem, but do they use it in its differential form?
The curl of gradient of scalar, also known as the Laplacian operator, is a mathematical operation that takes the gradient of a scalar field and then calculates the curl of the resulting vector. It is often used in physics and engineering to describe the behavior of fluid flow and electric fields.
The curl of gradient of scalar is used in many real-world applications, such as fluid dynamics, electromagnetism, and heat transfer. It is also commonly used in computer graphics to create realistic simulations of fluid flow and other physical phenomena.
The curl of gradient of scalar is significant because it helps us understand the behavior of vector fields in three-dimensional space. It is a useful tool for analyzing and solving complex problems in physics and engineering.
The curl of gradient of scalar is closely related to other vector operations, such as the divergence and gradient. The divergence of a vector field is the dot product of the gradient of the scalar field, while the curl is the cross product. These operations are all important in understanding the behavior of vector fields.
Yes, the curl of gradient of scalar can be negative. It depends on the direction and magnitude of the vector field being analyzed. A negative curl indicates that the vector field is rotating in a clockwise direction, while a positive curl indicates a counterclockwise rotation.