Example 1.6.1 The parametric equation r= (u+v)i + (u-v)j + 2uv k, where -0 < u...

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Example 1.6.1 The parametric equation r= (u+v)i + (u-v)j + 2uv k, where -∞0 < u < 00, -∞0 < < ∞o, gives a hyperbolic paraboloid. (It is the surface given by putting w = 0 in Example 1.1.3.) The two natural basis vectors are ₁₁ e₂ = e, So the quantities 9AB еA еg can be displayed as [9AB] = [2 ]. The inverse of this matrix is [AB], which we calculate to be 1+2² 2(1+2u²+20²) [AB] = [ 1+2u²+20² and Ər/du=i+j+2vk, dr/Ov=i-j+2uk. 2(1+20²) We can then give the dual basis vectors in terms of i, j, k by saying e"=e¹=g¹e=g¹¹e₁+g¹²e₂ -EV 1+2u²+2u2 4uv 4uv 2(1+2u²) 1+2u² 2(1+2u²+2v²) (i+j+ 2vk) + 1+2u²-2uv 2(1+2u²+20²) 2(1+2u²+2v²)+ i+ 1+2v² - 2uv 2(1 +2u² + 2v²) e" = e²=g²^cA=g²¹e₁ +9²2e₂ 1+2² 2(1+2u²+20²) -UV 2v k) + 1 + 2u² + 2v² (i −j + 2uk) 1+2u² +2uv (i+j+2vk) + V 1+2u²+2v² k 1+2v² 2(1+2u²+2u2) 1+ 2v² +2uv 2(1+2u²+20²) + j(i − j + 2uk) k. u 1+2u²+2v² Problems 2 1. Obtain the geodesic equations (using arc-length s as a parameter) for the hyperbolic paraboloid of Example 1.6.1. Deduce that all parametric curves are geodesics. Example 1.6.1 The parametric equation r= (u+v)i + (u-v)j + 2uv k, where -∞0 < u < 00, -∞0 < < ∞o, gives a hyperbolic paraboloid. (It is the surface given by putting w = 0 in Example 1.1.3.) The two natural basis vectors are ₁₁ e₂ = e, So the quantities 9AB еA еg can be displayed as [9AB] = [2 ]. The inverse of this matrix is [AB], which we calculate to be 1+2² 2(1+2u²+20²) [AB] = [ 1+2u²+20² and Ər/du=i+j+2vk, dr/Ov=i-j+2uk. 2(1+20²) We can then give the dual basis vectors in terms of i, j, k by saying e"=e¹=g¹e=g¹¹e₁+g¹²e₂ -EV 1+2u²+2u2 4uv 4uv 2(1+2u²) 1+2u² 2(1+2u²+2v²) (i+j+ 2vk) + 1+2u²-2uv 2(1+2u²+20²) 2(1+2u²+2v²)+ i+ 1+2v² - 2uv 2(1 +2u² + 2v²) e" = e²=g²^cA=g²¹e₁ +9²2e₂ 1+2² 2(1+2u²+20²) -UV 2v k) + 1 + 2u² + 2v² (i −j + 2uk) 1+2u² +2uv (i+j+2vk) + V 1+2u²+2v² k 1+2v² 2(1+2u²+2u2) 1+ 2v² +2uv 2(1+2u²+20²) + j(i − j + 2uk) k. u 1+2u²+2v² Problems 2 1. Obtain the geodesic equations (using arc-length s as a parameter) for the hyperbolic paraboloid of Example 1.6.1. Deduce that all parametric curves are geodesics.