What is meant by the angle between two surfaces at a point where they intersect? Determine the angle between the surfaces
at the point P = (1, 2, 1).
Along what ray emanating from the point P = (1, -1) does the function
increase most (respectively, least) rapidly?
What is its directional derivative at P in the direction of the first quadrant ray from P with slope 3/4 ?
The location at time t of a particle moving in space is given by:
| x = 2 cos t |
| y = 2 sin t |
| z = t |
the velocity.
the speed.
the unit tangent vector.
the acceleration.
the curvature.
the principal unit normal vector.
the binormal vector.
The graph of a function f of three variables is the set G_{f} of all solutions (x, y, z, w) of the equation f(x, y, z) - w = 0. Inasmuch as G_{f} is the set of all solutions of a single equation in 4 variables, it is an example of a 3-dimensional subset of 4-dimensional coordinate space. Such a set is called a hypersurface. In n-dimensional coordinate space a hypersurface is a subset of codimension 1, i.e., dimension n-1.
What vector with 4 coordinates would you expect to be normal in 4-space to G_{f}?
What hyperplane would you expect to be tangent to the graph of f(x, y, z) = z x^{3} + x y^{2} + y z at the point of G_{f} where (x, y, z)=(1, -1, 2) ?
Interpret the word hypersurface in n-dimensional space in the cases n = 1, 2, and 3.
What might be meant by the term unit hypersphere in 4-dimensional coordinate space?