Handbook of mathematics for engineers and scienteis61ts part ppt

388 DIFFERENTIAL GEOMETRY M M 0 u v Figure 9.20. The parametrized surface. 9.2.1-3. Tangent line to surface. A straight line is said to be tangent to a surface if it is tangent to a curve lying in this surface. Suppose that a surface is given in vector form (9.2.1.1) and a curve lying on it is parametrized by the parameter t. Then to each parameter value there corresponds a point of the curve, and the position of this point on the surface is specified by some values of the curvilinear coordinates u and v. Thus the curvilinear coordinates of points of a curve lying on a surface are functions of the parameter t. The system of equations u = u(t), v = v(t)(9.2.1.3) is called the intrinsic equations of the curve on the surface. The intrinsic equations completely characterize the curve if the vector equation of the surface is given, since the substitution of (9.2.1.3) into (9.2.1.1) results in the equation r = r[u(t), v(t)], (9.2.1.4) which is called the parametric equation of the curve. The differential of the position vector is equal to dr = r u du + r v dv,(9.2.1.5) where du = u  t (t) dt and dv = v  t (t) dt. The vectors r u and r v are called the coordinate vectors corresponding to the point whose curvilinear coordinates have been used in the computations. The coordinate vectors are tangent vectors to coordinate curves (Fig. 9.21). N r r v u Figure 9.21. Coordinate vectors. 9.2. THEORY OF SURFACES 389 Formula (9.2.1.5) shows that the direction vector of any tangent line to a surface at a given point is a linear combination of the coordinate vectors corresponding to this point; i.e., the tangent to a curve lies in the plane spanned by the vectors r u and r v at this point. The direction of the tangent to a curve on a surface at a point M is completely charac- terized by the ratio dv :du of differentials taken along this curve. 9.2.1-4. Tangent plane and normal. If all possible curves are drawn on a surface through a given regular point M 0 (r 0 )= M 0 (x 0 , y 0 , z 0 )=M(u 0 , v 0 ) of the surface, then their tangents at M 0 lie in the same plane, which is called the tangent plane to the surface at M 0 . The tangent plane can be defined as the limit position of the plane passing through three distinct points M 0 , M 1 ,andM 2 on the surface as M 1 → M 0 and M 2 → M 0 ;hereM 1 and M 2 should move along curves with distinct tangents at M 0 . The tangent plane at M 0 can be viewed as the plane passing through M 0 and perpen- dicular to the vector r u × r v ; i.e., it passes through the vectors r u and r v . Thus the tangent plane at M 0 , depending on the method for defi ning the surface, is given by one of the equations      x – x 0 y – y 0 z – z 0 x u y u z u x v y v z v      = 0,[(r – r 0 )r u r v ]=0, F x (x – x 0 )+F y (y – y 0 )+F z (z – z 0 )=0, z x (x – x 0 )+z y (y – y 0 )=z – z 0 , where all the derivatives are evaluated at the point M 0 (r 0 )=M 0 (x 0 , y 0 , z 0 )=M(u 0 , v 0 ). The straight line passing through M 0 and perpendicular to the tangent plane is called the normal to the surface at M 0 . The vector N = r u × r v /|r u × r v | is called the unit normal vector. The sense of the vector N is called the positive normal direction; the vector r u ,the vector r v , and the positive normal form a right triple. The equation of the normal, depending on the method for defining the surface, has one of the forms x – x 0    y u z u y v z v    = y – y 0    z u x u z v x v    = z – z 0    x u y u x v y v    , r = r 0 + λ(r u × r v )orr = r 0 + λN, x – x 0 F x = y – y 0 F y = z – z 0 F z , x – x 0 z x = y – y 0 z y = z – z 0 –1 , where all derivatives are evaluated at the point M 0 (r 0 )=M 0 (x 0 , y 0 , z 0 )=M(u 0 , v 0 ). Example 5. For the sphere given by the implicit equation x 2 + y 2 + z 2 – a 2 = 0, the tangent plane at the point M 0 (x 0 , y 0 , z 0 ) is given by the equation 2x 0 (x – x 0 )+2y 0 (y – y 0 )+2z 0 (z – z 0 )=0 or xx 0 + yy 0 + zz 0 = 0, and the normal is given by the equation x – x 0 2x 0 = y – y 0 2y 0 = z – z 0 2z 0 or x x 0 = y y 0 = z z 0 . 390 DIFFERENTIAL GEOMETRY 9.2.1-5. First quadratic form. If a surface is given parametrically or in vector form, M 1 (u, v) is an arbitrary point, and M 2 (u + du, v + dv) is a nearby point on the surface, then the length of the arc M 1 M 2 on the surface is approximately expressed in terms of the arc length differential or in terms of the linear surface element by the formula ds 2 = Edu 2 + 2Fdudv+ Gdv 2 ,(9.2.1.6) where the coefficients E, F ,andG are given by the formulas E = r 2 u = x 2 u + y 2 u + z 2 u , F = r u ⋅ r v = x u x v + y u y v + z u z v , G = r 2 v = x 2 v + y 2 v + z 2 v . The right-hand side of formula (9.2.1.6) is also called the first quadratic form of the surface given parametrically or in vector form; its coefficients E, F ,andG depend on the point on the surface. At each regular point on the surface corresponding to (real) coordinates u and v,thefirst quadratic form (0.2.1.12) is positive definite; i.e., E > 0, G > 0, EG – F 2 > 0. Example 6. For the sphere given by the equation r = a cos u sin vi + a sin u sin vj + a cos vk, the coefficients E, F ,andG are equal to E = a 2 sin 2 v, F = 0, G = a 2 , and the first quadratic form is ds 2 = a 2 (sin 2 vdu 2 + dv 2 ). For a surface given explicitly, the coefficients E, F ,andG are given by the formulas E = 1 + z 2 x , F = z x z y , G = 1 + z 2 y . The arc length of the curve u = u(t), v = v(t), t [t 0 , t 1 ], on the surface can be calculated by the formula L =  t 1 t 0 ds =  t 1 t 0  Eu 2 t + 2Fu t v t + Gv 2 t dt.(9.2.1.7) The angle γ between two curves (i.e., between their tangents) intersecting in a point M and having the direction vectors dr =(du, dv)andδr =(δu, δv) at this point (Fig. 9.22) can be calculated by the formula cos γ = dr δr |dr||δr| = Eduδu+ F (du δv + dv δu)+Gdvδv √ Edu 2 + 2Fdudv+ Gdv 2 √ Eδu 2 + 2Fδuδv+ Gδv 2 .(9.2.1.8) (The coefficients E, F ,andG are evaluated at point M.) dr δr γ Figure 9.22. The angle between two space curves. 9.2. THEORY OF SURFACES 391 In particular, the angle γ 1 between the coordinate curves u = const and v = const passing through a point M (u, v) is determined by the formulas cos γ 1 = F √ EG ,sinγ 1 = √ EG – F 2 √ EG . The coordinate lines are perpendicular if F = 0. The area of a domain U bounded by some curve on the surface can be calculated as the double integral S =  U dS =  U √ EG – F 2 du dv.(9.2.1.9) Thus if the coefficients E, F ,andG of the first quadratic form are known, then one can measure lengths, angles, and areas on the surface according to formulas (9.2.1.7), (9.2.1.8), and (9.2.1.9); i.e., the first quadratic form completely determines the intrinsic geometry of the surface (see Subsection 9.2.3 for details). To calculate surface areas in three-dimensional space, one can use the following theo- rems. T HEOREM 1. If a surface is given in the explicit form z = f(x, y) and a domain U on the surface is projected onto a domain V on the plane (x, y) ,then S =  V  1 + f 2 x + f 2 y dx dy. T HEOREM 2. If the surface is given implicitly ( F (x, y, z)=0 ) and a domain U on the surface is projected bijectively onto a domain V on the plane (x, y) ,then S =  V |grad F | |F z | dx dy, where |F z | = ∂F/∂z ≠ 0 for (x, y, z) lying in the domain U . THEOREM 3. If a surface is the parametric form r = r(u, v) or x = x(u, v) , y = y(u, v) , z = z(u, v) ,then S =  V |r u × r v | du dv. 9.2.1-6. Singular (conic) points of surface. A point M 0 (x 0 , y 0 , z 0 ) on a surface given implicitly, i.e., determined by the equation F (x, y)=0,issaidtobesingular (conic) if its coordinates satisfy the system of equa- tions F x (x 0 , y 0 , z 0 )=0, F y (x 0 , y 0 , z 0 )=0, F z (x 0 , y 0 , z 0 )=0, F(x 0 , y 0 , z 0 )=0. All tangents passing through a singular point M 0 (x 0 , y 0 , z 0 ) do not lie in the same plane but form a second-order cone defined by the equation F xx (x – x 0 )+F yy (y – y 0 )+F zz (z – z 0 )+2F xy (x – x 0 )(y – y 0 ) + 2F yz (y – y 0 )(z – z 0 )+2F zx (z – z 0 )(x – x 0 )=0. The derivatives are evaluated at the point M 0 (x 0 , y 0 , z 0 ); if all six second partial derivatives are simultaneously zero, then the singular point is of a more complicated type (the tangents form a cone of third or higher order). 392 DIFFERENTIAL GEOMETRY 9.2.2. Curvature of Curves on Surface 9.2.2-1. Normal curvature. Meusnier’s theorem. Of the plane sections of a surface, the planes containing the normal to the surface at a given point are said to be normal. In this case, there exists a unique normal section Γ 0 containing a given tangent to the curve Γ. M EUSNIER THEOREM. The radius of curvature at a given point of a curve Γ lying on a surface is equal to the radius of curvature of the normal section Γ 0 taken at the same point with the same tangent, multiplied by the cosine of the angle α between the osculating plane of the curve at this point and the plane of the normal section Γ 0 ; i.e., ρ = ρ N cos α‡ The normal curvature of a curve Γ at a point M (u, v)isdefined as k N = kn ⋅ N = r  ss ⋅ N =–r  s ⋅ N  s . The normal curvature is the curvature of the normal section. The geodesic curvature of a curve Γ at a point M(u, v)isdefined as k G = kr  s nN = r  s r  ss N. The geodesic curvature is the angular velocity of the tangent to the curve around the normal. The geodesic curvature is the curvature of the projection of the curve Γ onto the tangent plane. For any point u, v of the curve Γ given by equation (9.2.1.4) and lying on the surface, the curvature vector can be represented as a sum of two vectors, r  ss = kn = k G N × r  s + k N N,(9.2.2.1) where N is the unit normal vector to the surface. The first term on the right-hand side in (9.2.2.1) is called the geodesic (tangential) curvature vector, and the second term is called the normal curvature vector. The geodesic curvature vector lies in the tangent plane, and the normal curvature vector is normal to the surface. 9.2.2-2. Second quadratic form. Curvature of curve on surface. The quadratic differential form – dr ⋅ dN = Ldu 2 + 2Mdudv+ Ndv 2 , L =–r u ⋅ N u = r uu r u r v √ EG – F 2 , N =–r v ⋅ N v = r vv r u r v √ EG – F 2 , M =–r u ⋅ N v =–r v ⋅ N u = r uv r u r v √ EG – F 2 (all derivatives are evaluated at the point M (u, v)) is called the second quadratic form of the surface. 9.2. THEORY OF SURFACES 393 The coefficients L, N ,andM for surfaces given parametrically or implicitly can be calculated by the formulas L = 1 √ EG – F 2      x uu y uu z uu x u y u z u x v y v z v      = z xx  1 + z 2 x + z 2 y , M = 1 √ EG – F 2      x uv y uv z uv x u y u z u x v y v z v      = z xy  1 + z 2 x + z 2 y , N = 1 √ EG – F 2      x vv y vv z vv x u y u z u x v y v z v      = z yy  1 + z 2 x + z 2 y . The curvature k N of a normal section can be calculated by the formula k N =– dr ⋅ dN ds 2 = Ldu 2 + 2Mdudv+ Ndv 2 Edu 2 + 2Fdudv+ Gdv 2 . A point on the surface at which the curvature ρ N of a normal section takes the same value for any normal section (L : M : N = E : F : G)issaidtobeumbilical (circular).At each nonumbilical point, there are two normal sections called the principal normal sections. They are characterized by the maximum and minimum values k 1 and k 2 of the curvature ρ N , which are called the principal curvatures of the surface U at the point M(u, v). The planes of principal normal sections are mutually perpendicular. E ULER THEOREM. For a normal section at M(u, v) whose plane forms an angle θ with the plane of one of the principal normal sections, one has k N = k 1 cos 2 θ + k 2 sin 2 θ or k N = k 1 +(k 2 – k 1 )sin 2 θ. The quantities k 1 and k 2 are the roots of the characteristic equation    L – kE M – kF M – kF N – kG    = 0. The curves on a surface whose directions at each point coincide with the directions of the principal normal sections are called the curvature lines; their differential equation is      dv 2 –dv du du 2 EFG LMN      = 0. Asymptotic lines are defined to be the curves for which ρ N = 0 at each point. The asymptotic lines are determined by the differential equation Ldu 2 + 2Mdudv+ Ndv 2 = 0. 9.2.2-3. Mean and Gaussian curvatures. The symmetric functions H(u, v)= k 1 + k 2 2 and K(u, v)=k 1 k 2 394 DIFFERENTIAL GEOMETRY are called the mean and Gaussian (extrinsic) curvature, respectively, of the surface U at the point M(u, v). They are given by the formulas H(u, v)= EN – 2FM + LG 2(EG – F 2 ) , K(u, v)= LN – M 2 EG – F 2 .(9.2.2.2) The mean and Gaussian curvatures are related by the inequality H 2 – K = (k 1 + k 2 ) 2 4 – k 1 k 2 = (k 1 – k 2 ) 2 4 ≥ 0. The mean and Gaussian curvatures can be used to characterize the deviation of the surface from a plane. In particular, if H = 0 and K = 0 at all points of the surface, then the surface is a plane. Example 1. For a circular cylinder (of radius a), H = a 2 , K = 0. If a surface is represented by the equation z = f (x, y), then the mean and Gaussian curvature can be determined by the formulas H = r(1 + q 2 )–2pqs + t(1 + p 2 ) 2  (1 + p 2 + q 2 ) 3 , K = rt – s 2 (1 + p 2 + q 2 ) 3 , where the following notation is used: p = z x , q = z y , r = z xx , s = z xy , t = z yy , h =  1 + p 2 + q 2 . The surfaces for which the mean curvature H is zero at all points are said to be minimal. The surfaces for which the Gaussian curvature K is constant at all points are called surfaces of constant curvature. 9.2.2-4. Classification of points on surface. The points of a surface can be classified according to the values of the Gaussian curvature: 1. A point M at which K = k 1 k 2 > 0 (the principal normal sections are convex in the same direction from the tangent plane; example: any point of an ellipsoid) is called an elliptic point; the analytic criterion for this case is LN – M 2 > 0. In the special case k 1 = k 2 , the point is umbilical (circular): R = const for all normal sections at this point. 2. A point M at which K = k 1 k 2 < 0 (the principal normal sections are convex in opposite directions; the surface intersects the tangent plane and has a saddle character; example: any point of a one-sheeted hyperboloid) is called a hyperbolic (saddle) point; the analytic criterion for this case is LN – M 2 < 0. 3. A point M at which K = k 1 k 2 = 0 (one principal normal section has an inflection point or is a straight line; example: any point of a cylinder) is called a parabolic point;the analytic criterion for this case is LN – M 2 = 0. Any umbilical point (k 1 = k 2 ) is either elliptic or parabolic. . y 2 v + z 2 v . The right-hand side of formula (9.2.1.6) is also called the first quadratic form of the surface given parametrically or in vector form; its coefficients E, F ,andG depend on the point. ,andG of the first quadratic form are known, then one can measure lengths, angles, and areas on the surface according to formulas (9.2.1.7), (9.2.1.8), and (9.2.1.9); i.e., the first quadratic form. specified by some values of the curvilinear coordinates u and v. Thus the curvilinear coordinates of points of a curve lying on a surface are functions of the parameter t. The system of equations u =