- #1
Dave-o
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Homework Statement
∇ . r = 3, ∇ x r = 0
Homework Equations
The Attempt at a Solution
So far I've gotten up to ∇(∇^2 r)
∇(∇r(hat)/r) is the gradient of the gradient of the position vector, divided by the magnitude of the position vector. It is evaluated by taking the derivative of the position vector twice, and then dividing by the magnitude of the position vector.
Evaluating ∇(∇r(hat)/r) allows us to calculate the curvature of a surface at a specific point, which is important in understanding the behavior of objects in that location.
Yes, ∇(∇r(hat)/r) can be simplified using vector calculus identities and properties. For example, it can be rewritten as (∇^2)r(hat) - (∇r(hat) • ∇r(hat))/r.
The physical interpretation of ∇(∇r(hat)/r) is the rate of change of the unit normal vector (or surface normal) with respect to the position vector. It describes how the orientation of a surface changes as we move along it.
Evaluating ∇(∇r(hat)/r) is useful in many fields, including physics, engineering, and mathematics. It is commonly used in studying the behavior of electric and magnetic fields, as well as in fluid dynamics and elasticity. It is also important in understanding the geometry of surfaces in calculus and differential geometry.