
The earth's spherical coordinate system
The projection of the dotted line on the x axis is:
(1.2)
While the projection of the dotted line on the y axis is:
(1.3)
In our case the angles are:
And R=6367km
Thus, Beijing Cartesian coordinates (in km) are:
Beijing Cartesian

Radius of Curvature
Therefore the radius of the curvature is
=2 .
To sketch the curve , let us observe that its projection to the plane is a circle with the radius and the center at (0,0,0).

Stereographic Projection, Riemann Sphere and Mapping
Well let's pick a point on the x axis: x=a
Now we move this point through space to form a line in the x y plane, every time y increases by 1, x gets incremented by a fixed amount, say b, so: x=a+by
Now we move this line through space to form

10 Calculus Problems
x+2y+4z = 8
Interception on x axis: y=0,z=0, x = 8
Interception with y axis: x=0,z=0, y = 4
Interception with zaxis: x=0,y=0, z = 2
therefore,
three well known points are: (8,0,0), (0,4,0), (0,0,2)
See the attached file.
2.)

Stereographic projection on complex plane
After the stereographic projection, it is mapped to the point z=x+iy in the complex plane, where x=x_1/(1x_3) and y=x_2/(1x_3). We know (x_1)^2+(x_2)^2+(x_3)^2=1 since the point (x_1,x_2,x_3) is on the sphere S.

Indexed Families of Sets
b) The family of all circles in pi of radius 1 whose center is on the yaxis.
c) The family of all circles in pi of radius 1.
d) The family of all lines in pi with slope 6.

Formula of Circle
see that the x coordinate of the point on the circle can be substituted into the "a" and the y coordinate can be substituted into the "b", and the radius(r) can be substituted into the "c" of the Pythagorean Theorem.

BolzanoWeierstrauss
Prove that the set of open disks in the xy plane with center (x,x) and radius x > 0, x rational, is a countable covering of the set {(x,y): x > 0, y > 0} Let A={(x,y),x>0,y>0} and U_x={(u,v):(ux)^2+(vx)^2<=x^2, x is a rational number}.

Vector integral  stokes therom
The normal to the plane is a vector parallel to (1,0,1), the coefficients of x, y, z from plane equation.