The Gradient Theorem: Understanding the Physical Interpretation and Intuition

[member 428835]

i am not sure if this post should be under calculus or not, but i think i'll get a more "complete" answer here. at any rate, I'm wondering if anyone can clarify the intuition behind the gradient theorem: $$\iiint\limits_V \nabla \psi dV=\iint\limits_S \psi \vec{dS}$$ by intuition, i refer to a physical interpretation. for instance, i understand the divergence theorem states (rate of expansion of a vector field) = (rate the vector field leaks out of the edges). but this extended gradient theorem is more difficult. please help! i doubt i need to clarify this notation, but please let me know if I've been ambiguous.

thanks!

 

[Chestermiller]

i am not sure if this post should be under calculus or not, but i think i'll get a more "complete" answer here. at any rate, I'm wondering if anyone can clarify the intuition behind the gradient theorem: $$\iiint\limits_V \nabla \psi dV=\iint\limits_S \psi \vec{dS}$$ by intuition, i refer to a physical interpretation. for instance, i understand the divergence theorem states (rate of expansion of a vector field) = (rate the vector field leaks out of the edges). but this extended gradient theorem is more difficult. please help! i doubt i need to clarify this notation, but please let me know if I've been ambiguous.

thanks!

This is just the 3D version of $\int_a^b{\frac{df}{dx}dx}=f(b)-f(a)$

 

[member 428835]

yes, i do realize this, but I am a little shakey on the intuition. the slope vectors added up make sense to check the difference of endpoints, but the 2 and 3 dimensions are not as obvious.

 

[Mandelbroth]

i am not sure if this post should be under calculus or not, but i think i'll get a more "complete" answer here. at any rate, I'm wondering if anyone can clarify the intuition behind the gradient theorem: $$\iiint\limits_V \nabla \psi dV=\iint\limits_S \psi \vec{dS}$$ by intuition, i refer to a physical interpretation. for instance, i understand the divergence theorem states (rate of expansion of a vector field) = (rate the vector field leaks out of the edges). but this extended gradient theorem is more difficult. please help! i doubt i need to clarify this notation, but please let me know if I've been ambiguous.

thanks!

When you post in the differential geometry part of the forum, you get a differential geometry answer.

Note first, though, that your "gradient theorem" is actually just a form of the Divergence Theorem. It turns out that all these integral theorems that you learn from multivariate calculus are actually results of a single theorem called the generalized Stokes' Theorem from differential geometry.

The intuition for your problem can be seen in 2 dimensions. From a topological standpoint, we can integrate over chains, and in the two dimensional case we'll look at a 2-chain and its boundary and see what integration over the two might be like. Look at the pictures below (Ignore the rainbows. Life in mathland is just that happy).

The first shows the oriented boundary of a region in two dimensional space. The second shows finer and finer oriented tilings of the region. Note that the interior arrows of each tiling go in opposite directions from their neighbors, "cancelling" each other out. The integral over the 2-chain would be equal to, in a sense, the contribution of the boundary.

Does this help?

 

[blue_raver22]

The gradient theorem is a fundamental concept in vector calculus that relates the flow of a vector field through a surface to the behavior of the field within a volume. It can be understood in terms of the physical interpretation of the gradient, which represents the direction and magnitude of the steepest increase in a scalar field. In other words, the gradient points in the direction of greatest change in the scalar field.

In the context of the gradient theorem, the scalar field \psi represents a physical quantity, such as temperature or pressure, that is changing within a volume V. The surface S represents the boundary of this volume, and the vector \vec{dS} represents the direction and magnitude of the flow of the scalar field through this boundary.

The gradient theorem states that the total amount of change in the scalar field within the volume V is equal to the net flow of the field through the boundary S. This can be understood intuitively as follows: if the scalar field is increasing within the volume, the gradient will be pointing in the direction of this increase. This means that the flow of the field through the boundary will be directed outward, as the field is expanding. Similarly, if the scalar field is decreasing within the volume, the gradient will be pointing in the opposite direction, and the flow of the field through the boundary will be directed inward, as the field is contracting.

In this way, the gradient theorem provides a physical interpretation for the relationship between the behavior of a scalar field within a volume and the flow of the field through its boundary. It can be thought of as a generalized version of the divergence theorem, which relates the divergence of a vector field to its flow through a closed surface. The gradient theorem extends this concept to a more general setting, where the scalar field may be changing within the volume and the surface may be open.

Overall, the gradient theorem is a powerful tool for understanding the behavior of vector fields and their relationship to scalar fields. It allows us to make connections between seemingly disparate concepts and provides a deeper understanding of the physical processes at play. I hope this explanation helps clarify the intuition behind the gradient theorem.