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So, I read somewhere that the Jacobian determinant of a transformation determines the local volume change.

Say I am in 3D space and I have the following relationship:

F(x', y', z') = F(x, y, z) + T(x, y, z)

The LHS gives the new position and the RHS is the old position + the displacement vector.

So, the gradient is:

Identity + Jacobian.

Now, my question is: what is this Jacobian of the displacement field telling me? From what I read, it sort of represents ascalingof the local volume. However, I am struggling to see that.

Also, is the identity matrix in the gradient formulation important to calculate this scaling factor or just looking at the Jacobian determinant of the displacement field suffice?? From what I have been reading on the net so far, the displacement field and its jacobian tells us everything about the volume change...

I have been struggling for a while to visualize and understand this Jacobian stuff but with no luck...

I would be really grateful for your help.

Many thanks,

Luca

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# Jacobian determinant and volume change

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