For positive constants a,b,c>0, consider the ellipsoid Sa,b,cR3 defined by Sa,b,c={(x,y,z)R3a2x2+b2y2+c2z2=1}. By the Regular Value Theorem, Sa,b,cR3 is a smooth surface. For pSa,b,c, let KpR be the Gaussian curvature of Sa,b,c at p. Note that subsurface Sa,b,c+={(x,y,z)Sa,b,cz0} can be written as the graph of the function f(x,y)=c1a2x2b2y2, where the domain of f is the set D={(x,y)R2a2x2+b2y21}. 4.1. Using your answer to Question 1.1(1), calculate the Gaussian curvature function K:Sa,b,cR,pKp. Hint: You can freely use the following facts: the map K:Sa,b,cR is a continuous function, the diffeomorphism F:Sa,b,cSa,b,c,(x,y,z)(x,y,z) is an isometry and isometries preserve Gaussian curvature. Hint: This question requires non-trivial calculation. You might want to write =a21,=b21 and h=1a2x2b2y2 to save some ink. 4.2. Using your answer to Part 1, deduce that Kp>0 for all pSa,b,c and that K is a constant function if and only if a=b=c. 4.3. Using your answer to Part 2, deduce that Sa,b,c is isometric to a sphere Sr2 if and only if a=b=c=r. For positive constants a,b,c>0, consider the ellipsoid Sa,b,cR3 defined by Sa,b,c={(x,y,z)R3a2x2+b2y2+c2z2=1}. By the Regular Value Theorem, Sa,b,cR3 is a smooth surface. For pSa,b,c, let KpR be the Gaussian curvature of Sa,b,c at p. Note that subsurface Sa,b,c+={(x,y,z)Sa,b,cz0} can be written as the graph of the function f(x,y)=c1a2x2b2y2, where the domain of f is the set D={(x,y)R2a2x2+b2y21}. 4.1. Using your answer to Question 1.1(1), calculate the Gaussian curvature function K:Sa,b,cR,pKp. Hint: You can freely use the following facts: the map K:Sa,b,cR is a continuous function, the diffeomorphism F:Sa,b,cSa,b,c,(x,y,z)(x,y,z) is an isometry and isometries preserve Gaussian curvature. Hint: This question requires non-trivial calculation. You might want to write =a21,=b21 and h=1a2x2b2y2 to save some ink. 4.2. Using your answer to Part 1, deduce that Kp>0 for all pSa,b,c and that K is a constant function if and only if a=b=c. 4.3. Using your answer to Part 2, deduce that Sa,b,c is isometric to a sphere Sr2 if and only if a=b=c=r