The directional derivative is given by
Dufx, y, z) = ∇fx, y, z⋅u, where u is a unit
vector in the direction of the outward normal to the sphere
x2 + y2 + z2 = 9. The vector
∇x2 + y2 + z2 =
is normal to the sphere
x2 + y2 + z2 = 9, and at
2, 2, 1
this normal is
4, 4, 2. A unit vector in the same direction
is given by
u = ÷ = =
Hence,
∇fx, y, z⋅u evaluated at (2, 2, 1) is
- .BITMAPSETProbSolvHint0.2006in0.243in0inq1
2.
A normal vector is given by
∇ + x2 + y2 - z
= x + 3x,y + 3y, - 1
= x, y, - 1
= x1 + 3x2 +3y2, y1 + 3x2 +3y2, -
Hence, any scalar multiple of BITMAPSETProbSolvHint0.2006in0.243in0inq2
x1 + 3x2 +3y2, y1 + 3x2 +3y2, -
is also normal to the given surface.
1.
Evaluate the expression
∇. Then delete the first two rows of the vector,
because m and M are constant parameters. BITMAPSETProbSolvHint0.2006in0.243in0inq3
∇ 8pt
= - mx, - my, - mz 8pt
= - x, y, z
This gives the Newtonian gravitational force between two objects of masses m and M, with one object at the origin and the other at the point
x, y, z.