Chapter 1 Exercises
1.1 (a)
Direction of maximum rate of change is along the gradient vector.
Find the gradient vector and plug in the point to find the direction of most rapid rate of change at that point.
The elements of a basis are usually chosen to be unit vectors. Every vector in the space may be written as a linear combination of unit vectors, with the components of each being given by direction cosines
1.1 (b)
To find the maximum rate of change, find the magnitude of the gradient vector
1.2 (a)
In vector calculus, the divergence theorem, also known as Gauss' theorem, Ostrogradsky's theorem, or Gauss-Ostrogradsky theorem is a result that relates the outward flow of a vector field on a surface to the behaviour of the vector field inside the surface.
More precisely, the divergence theorem states that the flux of a vector field on a surface is equal to the triple integral of the divergence on the region inside the surface. Intuitively, it states that the sum of all sources minus the sum of all sinks gives the net flow out of a region.
The divergence theorem is an important result for the mathematics of physics, in particular in electrostatics and fluid dynamics.