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The shortest path between two points on a curved surface, such as the surface of a sphere is called a geodesic. To find a geodestic, one has first to set up an integral that gives the length of a path on the surface in question. Use sperical polar coordinates (r,theat,phi) to show that the length of a path joining two points on a shere of radius R is L=R*integral(from theta_1 to theta_2)sqrt(1+sin^2(theta)*(phi_prime(theta))^2)*d(theta)

if (theta_1,phi_1) and (theta_2,phi_2) specify two points and we assume that the path is expressed as phi=phi(theta).

I know how to do this problem if it were just (x,y) it would be L=integral(from x_1 to x_2) ds, where ds= sqrt(dx^2+dy^2)= sqrt(1+y_prime(x)^2) dx

but im getting confused on how to impliment the spherical polar cords. For example x=r*sin(phi)*cos(theta), but then i dont know what dx would be because im not sure what to differantiat?

thanks for the help.