(Solved): Question 1 (25 marks) - Vector valued functions \& curvature Consider the vector valued function F: ...

Question 1 (25 marks) - Vector valued functions \& curvature Consider the vector valued function $$F:{\mathbf{R}}^2?{\mathbf{R}}^3$$ given by: $$F(u,v)=(2\sin (u)\sin (v),2\sin (u)\cos (v),2\cos (u)).$$ In the following, we shall write $$f_1,f_2,f_3:{\mathrm{R}}^2?\mathrm{R}$$ for the three component functions of $$F$$, meaning $$F=\left(f_1,f_2,f_3\right)$$. (i) (5 marks) Show that any point, $$x?F\left({\mathbb{R}}^2\right)$$, in the image of $${\mathbb{R}}^2$$ under $$F$$ lies on the 2-sphere; the 2-sphere consists of all points in $${\mathrm{R}}^3$$ of Euclidean distance 2 from the origin, $$0?{\mathbb{R}}^3$$. (ii) (5 marks) Determine whether $$F$$ is differentiable and justify your answer. If affirmative, write down its (total) derivative $$DF(u,?)$$. (iii) $$(7.5$$ marks) Next compute the following vector-valued functions of first and second order partial derivatives: 1 (iv) (7.5 marks) Viewing F as a parameterisation of the 2-sphere, compute the Gaußian curvature $$K$$ of this surface, defined by the formula: $$K=\frac{\det \left(F_{vu}{F}_v{F}_v\right)\det \left(F_{vw}{F}_v{F}_v\right)?\det {\left(F_{vv}{F}_u{F}_v\right)}^2}{{\left({?F_v?}^2{?F_v?}^2?{\left(F_v?F_v\right)}^2\right)}^2}$$ ( 5 marks). Do you observe anything striking about the curvature of the 2-sphere and, if so, is this expected? ( $$2.5$$ marks)