answer:To find the sectional curvature of the surface z = x^2 + y^2 at point P(1,1,2), we first need to find the first and second fundamental forms of the surface. Let's parameterize the surface as follows: S(u, v) = (u, v, u^2 + v^2) Now, we find the first partial derivatives: S_u = (1, 0, 2u) S_v = (0, 1, 2v) Next, we find the second partial derivatives: S_uu = (0, 0, 2) S_uv = (0, 0, 0) S_vv = (0, 0, 2) Now, we find the coefficients of the first fundamental form: E = S_u • S_u = 1 + 4u^2 F = S_u • S_v = 2uv G = S_v • S_v = 1 + 4v^2 At point P(1,1,2), we have: E = 1 + 4(1)^2 = 5 F = 2(1)(1) = 2 G = 1 + 4(1)^2 = 5 Now, we find the coefficients of the second fundamental form: L = S_uu • N = 2 / sqrt(1 + 4u^2 + 4v^2) M = S_uv • N = 0 N = S_vv • N = 2 / sqrt(1 + 4u^2 + 4v^2) At point P(1,1,2), we have: L = 2 / sqrt(1 + 4(1)^2 + 4(1)^2) = 2 / sqrt(17) M = 0 N = 2 / sqrt(1 + 4(1)^2 + 4(1)^2) = 2 / sqrt(17) Finally, we compute the Gaussian curvature K and the mean curvature H: K = (LN - M^2) / (EG - F^2) H = (LE + GN - 2FM) / (2(EG - F^2)) At point P(1,1,2), we have: K = (2/17 * 2/17 - 0^2) / (5*5 - 2^2) = 4/289 / 21 = 4 / (289 * 21) H = (2/17 * 5 + 2/17 * 5 - 2*0) / (2(5*5 - 2^2)) = 20/17 / 42 = 10 / (17 * 21) The sectional curvature of the surface at point P(1,1,2) is the Gaussian curvature K, which is: K = 4 / (289 * 21)

answer:To find the sectional curvature, we first need to compute the first and second fundamental forms of the surface. Given the parametric equations x(u,v) = u^2 + v^2, y(u,v) = uv, z(u,v) = u - v^2, we can find the partial derivatives with respect to u and v: x_u = 2u, x_v = 2v y_u = v, y_v = u z_u = 1, z_v = -2v These partial derivatives represent the tangent vectors T_u and T_v: T_u = (2u, v, 1) T_v = (2v, u, -2v) Now, we can find the normal vector N by taking the cross product of T_u and T_v: N = T_u × T_v = (2u*v + 2v^2, 2u^2 + 2v^2, 2u^2 - 2u*v) Now, we can compute the coefficients of the first fundamental form: E = T_u • T_u = (2u)^2 + v^2 + 1^2 = 4u^2 + v^2 + 1 F = T_u • T_v = 2u*2v + v*u + 1*(-2v) = 2uv - 2v G = T_v • T_v = (2v)^2 + u^2 + (-2v)^2 = 4v^2 + u^2 + 4v^2 = u^2 + 8v^2 Next, we compute the second derivatives: x_uu = 2, x_uv = 0, x_vv = 2 y_uu = 0, y_uv = 1, y_vv = 0 z_uu = 0, z_uv = 0, z_vv = -2 Now, we can compute the coefficients of the second fundamental form: L = N • (x_uu, y_uu, z_uu) = (2u*v + 2v^2)(2) + (2u^2 + 2v^2)(0) + (2u^2 - 2u*v)(0) = 4u*v + 4v^2 M = N • (x_uv, y_uv, z_uv) = (2u*v + 2v^2)(0) + (2u^2 + 2v^2)(1) + (2u^2 - 2u*v)(0) = 2u^2 + 2v^2 N = N • (x_vv, y_vv, z_vv) = (2u*v + 2v^2)(2) + (2u^2 + 2v^2)(0) + (2u^2 - 2u*v)(-2) = 4u*v - 4u^2 + 4v^2 Now, we need to find the values of these coefficients at the point (1,1,0). Since x(1,1) = 1^2 + 1^2 = 2, y(1,1) = 1*1 = 1, and z(1,1) = 1 - 1^2 = 0, we have u = 1 and v = 1. E(1,1) = 4 + 1 + 1 = 6 F(1,1) = 2 - 2 = 0 G(1,1) = 1 + 8 = 9 L(1,1) = 4 + 4 = 8 M(1,1) = 2 + 2 = 4 N(1,1) = 4 - 4 + 4 = 4 Finally, we can compute the sectional curvature K: K = (LN - M^2) / (EG - F^2) = (8*4 - 4^2) / (6*9 - 0^2) = (32 - 16) / (54) = 16 / 54 = 8 / 27 So, the sectional curvature of the surface at the point (1,1,0) along the curve with tangent vector (-1,1,0) and normal vector (0,0,1) is 8/27.

answer:To compute the sectional curvature K(v,w) at the point (1,1,2), we first need to find the tangent vectors v and w, and then compute the Gaussian curvature K at the point (1,1,2) using the formula: K(v,w) = (R(v,w)w,v) / (|v|^2 * |w|^2 - (v,w)^2) where R(v,w) is the Riemann curvature tensor. 1. Find the tangent vectors v and w: The surface S is given by the equation z = x^2 + y^2. The gradient of S at the point (1,1,2) is: ∇S(1,1,2) = (2x, 2y, -1) = (2, 2, -1) The plane z = 2 has the normal vector n = (0, 0, 1). Now, we find the tangent vector v to the curve of intersection of S with the plane z = 2. This vector is orthogonal to both the gradient of S and the normal vector of the plane: v = ∇S × n = (2, 2, -1) × (0, 0, 1) = (2, -2, 0) To find the tangent vector w to a meridian of S passing through the point (1,1,2), we can take the cross product of the gradient of S and the tangent vector v: w = ∇S × v = (2, 2, -1) × (2, -2, 0) = (2, 2, 8) Now, we need to normalize the vectors v and w: |v| = sqrt(2^2 + (-2)^2 + 0^2) = 2 * sqrt(2) |w| = sqrt(2^2 + 2^2 + 8^2) = 2 * sqrt(21) v_normalized = (1/sqrt(2), -1/sqrt(2), 0) w_normalized = (1/sqrt(21), 1/sqrt(21), 4/sqrt(21)) 2. Compute the Gaussian curvature K at the point (1,1,2): The first and second fundamental forms of the surface S are given by: I = (E, F; F, G) = ((1 + 4x^2), 4xy; 4xy, (1 + 4y^2)) II = (L, M; M, N) = (2x, 2y; 2y, 2x) At the point (1,1,2), we have: I = (5, 4; 4, 5) II = (2, 2; 2, 2) Now, we compute the Gaussian curvature K using the formula: K = (EG - F^2) / (EL - M^2) = (5*2 - 4^2) / (2*2 - 2^2) = (10 - 16) / (4 - 4) = -6 / 0 Since the denominator is zero, the Gaussian curvature K is undefined at the point (1,1,2). Therefore, the sectional curvature K(v,w) is also undefined at this point.

answer:To find the sectional curvature of the surface z = x^2 + y^2 at the point (1,1,2), we first need to find the first and second fundamental forms of the surface. 1. Parametrization of the surface: Let's parametrize the surface as follows: r(u,v) = (u, v, u^2 + v^2) 2. First fundamental form: We need to find the tangent vectors of the surface with respect to u and v: ru = dr/du = (1, 0, 2u) rv = dr/dv = (0, 1, 2v) Now, we can find the coefficients of the first fundamental form: E = <ru, ru> = 1 + 4u^2 F = <ru, rv> = 4uv G = <rv, rv> = 1 + 4v^2 3. Second fundamental form: We need to find the normal vector of the surface: N = ru × rv = (-2u, -2v, 1) Now, we normalize the normal vector: |N| = sqrt((-2u)^2 + (-2v)^2 + 1^2) = sqrt(4u^2 + 4v^2 + 1) n = N/|N| = (-2u, -2v, 1) / sqrt(4u^2 + 4v^2 + 1) Next, we find the second derivatives of r with respect to u and v: ruu = d^2r/du^2 = (0, 0, 2) ruv = d^2r/dudv = (0, 0, 0) rvv = d^2r/dv^2 = (0, 0, 2) Now, we can find the coefficients of the second fundamental form: L = <ruu, n> = (2/sqrt(4u^2 + 4v^2 + 1)) M = <ruv, n> = 0 N = <rvv, n> = (2/sqrt(4u^2 + 4v^2 + 1)) 4. Sectional curvature: The sectional curvature K is given by the following formula: K = (LN - M^2) / (EG - F^2) At the point (1,1,2), we have u = 1 and v = 1. So, we can find the values of E, F, G, L, M, and N at this point: E = 1 + 4(1)^2 = 5 F = 4(1)(1) = 4 G = 1 + 4(1)^2 = 5 L = 2/sqrt(4(1)^2 + 4(1)^2 + 1) = 2/sqrt(9) = 2/3 M = 0 N = 2/sqrt(4(1)^2 + 4(1)^2 + 1) = 2/sqrt(9) = 2/3 Now, we can find the sectional curvature K: K = (LN - M^2) / (EG - F^2) = ((2/3)(2/3) - 0^2) / (5*5 - 4^2) = (4/9) / (25 - 16) = (4/9) / 9 = 4/81 The sectional curvature of the surface z = x^2 + y^2 at the point (1,1,2) is 4/81.