Handbook of mathematics for engineers and scienteis61ts part ppt

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 p

M0

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

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

which is called the parametric equation of the curve.

The differential of the position vector is equal to

dr = ru du+ rv dv, (9.2.1.5)

where du = u  t (t) dt and dv = v  t (t) dt The vectors r u and rv 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.

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 ru and rvat 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 M0(r0) =

M0(x0, y0, z0) = M (u0, v0) of the surface, then their tangents at M0lie in the same plane,

which is called the tangent plane to the surface at M0 The tangent plane can be defined

as the limit position of the plane passing through three distinct points M0, M1, and M2on

the surface as M1→ M0and M2→ M0; here M1and M2should move along curves with

distinct tangents at M0

The tangent plane at M0 can be viewed as the plane passing through M0 and perpen-dicular to the vector

r ×rv;

i.e., it passes through the vectors ru and rv Thus the tangent plane at M0, depending on the method for defining the surface, is given by one of the equations







x – x0 y – y0 z – z0





=0, [(r – r0)ru v] =0,

F x (x – x0) + F y (y – y0) + F z (z – z0) =0, z x (x – x0) + z y (y – y0) = z – z0,

where all the derivatives are evaluated at the point M0(r0) = M0(x0, y0, z0) = M (u0, v0)

The straight line passing through M0 and perpendicular to the tangent plane is called

the normal to the surface at M0 The vector N = ru×rv /|r ×rv|is called the unit normal

vector The sense of the vector N is called the positive normal direction; the vector r u, the

vector rv, 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 – x0



y y u z u

v z v



 =

y – y0



z z u x u

v x v



 =

z – z0



x x u y u

v y v



, r = r0+ λ(r u×rv) or r = r0+ λN,

x – x0

F x =

y – y0

F y =

z – z0

F z ,

x – x0

z x =

y – y0

z y =

z – z0

–1 ,

where all derivatives are evaluated at the point M0(r0) = M0(x0, y0, z0) = M (u0, v0)

Example 5 For the sphere given by the implicit equation

x2+ y2+ z2– a2= 0 ,

the tangent plane at the point M0(x0, y0, z0) is given by the equation

2x0(x – x0) + 2y0(y – y0) + 2z0(z – z0) = 0 or xx0+ yy0+ zz0= 0 , and the normal is given by the equation

x – x0

2x0 = y2– y0 y0 = z2– z0 z0 or x

x0 = y

y0 = z

z0.

9.2.1-5 First quadratic form.

If a surface is given parametrically or in vector form, M1(u, v) is an arbitrary point, and

M2(u + du, v + dv) is a nearby point on the surface, then the length of the arc M1M2on the

surface is approximately expressed in terms of the arc length differential or in terms of the linear surface element by the formula

ds2= E du2+2F du dv + G dv2, (9.2.1.6)

where the coefficients E, F , and G are given by the formulas

E= r2u = x2u + y2u + z2u,

F = ru⋅rv = x u x v + y u y v + z u z v,

G= r2v = x2v + y v2+ z v2

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 , and G depend on the point on the surface At each regular point on the surface corresponding to (real) coordinates u and

v, the first quadratic form (0.2.1.12) is positive definite; i.e.,

E >0, G>0, EG – F2>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 , and G are equal to

E = a2sin2v, F = 0 , G = a2, and the first quadratic form is

ds2= a2(sin2v du2+ dv2).

For a surface given explicitly, the coefficients E, F , and G are given by the formulas

E =1+ z x2, F = z x z y, G=1+ z y2

The arc length of the curve u = u(t), v = v(t), t[t0, t1], on the surface can be calculated

by the formula

L=

 t1

t0 ds=

 t1

t0



Eu2

t+2F u t v t + Gv t2dt. (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) and δr = (δu, δv) at this point (Fig 9.22) can

be calculated by the formula

cos γ = d r δr

|dr||δr| =

E du δu + F (du δv + dv δu) + G dv δv

√

E du2+2F du dv + G dv2√

E δu2+2F δu δv + G δv2. (9.2.1.8)

(The coefficients E, F , and G are evaluated at point M )

dr

δr

γ

Figure 9.22 The angle between two space curves.

In particular, the angle γ1between the coordinate curves u = const and v = const passing through a point M (u, v) is determined by the formulas

cos γ1= √ F

EG, sin γ1 =

√

EG – F2

√

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 – F2du dv (9.2.1.9)

Thus if the coefficients E, F , and G 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

THEOREM1 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 x2+ f y2dx dy

THEOREM2 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 ≠ 0for (x, y, z) lying in the domain U

THEOREM3 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 ×rv|du dv 9.2.1-6 Singular (conic) points of surface

A point M0(x0, y0, z0) on a surface given implicitly, i.e., determined by the equation

F (x, y) = 0, is said to be singular (conic) if its coordinates satisfy the system of

equa-tions

F x (x0, y0, z0) =0, F y (x0, y0, z0) =0, F z (x0, y0, z0) =0, F (x0, y0, z0) =0

All tangents passing through a singular point M0(x0, y0, z0) do not lie in the same plane but form a second-order cone defined by the equation

F xx (x – x0) + F yy (y – y0) + F zz (z – z0) +2F xy (x – x0)(y – y0)

+2F yz (y – y0)(z – z0) +2F zx (z – z0)(x – x0) =0

The derivatives are evaluated at the point M0(x0, y0, z0); 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)

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 sectionΓ0containing

a given tangent to the curveΓ

MEUSNIER THEOREM The radius of curvature at a given point of a curveΓ lying on a surface is equal to the radius of curvature of the normal sectionΓ0taken 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 sectionΓ0; i.e.,

ρ = ρ N cos α‡

The normal curvature of a curve Γ at a point M(u, v) is defined 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) is defined as

k G = kr  snN = r srssN.

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 curveΓ onto the tangent plane

For any point u, v of the curveΓ given by equation (9.2.1.4) and lying on the surface, the curvature vector can be represented as a sum of two vectors,

rss = kn = k GN×r s + k NN, (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⋅d N = L du2+2M du dv + N dv2,

L= –ru⋅Nu= √ruur rv

EG – F2, N = –rv⋅Nv = √rvvr rv

EG – F2,

M = –ru⋅Nv = –rv⋅Nu= √ruvr rv

EG – F2 (all derivatives are evaluated at the point M (u, v)) is called the second quadratic form of the surface.

The coefficients L, N , and M for surfaces given parametrically or implicitly can be

calculated by the formulas

√

EG – F2







x uu y uu z uu

x u y u z u

x v y v z v





= z xx



1+ z x2+ z y2,

EG – F2







x uv y uv z uv

x u y u z u

x v y v z v





= z xy



1+ z x2+ z2,

EG – F2







x vv y vv z vv

x u y u z u

x v y v z v





= z yy



1+ z x2+ z2

The curvature k N of a normal section can be calculated by the formula

k N = –dr⋅dN

ds2 =

L du2+2M du dv + N dv2

E du2+2F du dv + G dv2 .

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) is said to be umbilical (circular) At each nonumbilical point, there are two normal sections called the principal normal sections They are characterized by the maximum and minimum values k1and k2of 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

EULER 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 = k1cos2θ + k2sin2θ or k N = k1+ (k2– k1) sin2θ

The quantities k1and k2are the roots of the characteristic equation



M L – kE – kF M N – kG – kF =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







dv2 –dv du du2





=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

L du2+2M du dv + N dv2 =0

9.2.2-3 Mean and Gaussian curvatures

The symmetric functions

H (u, v) = k1+ k2

2 and K (u, v) = k1k2

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–2F M + LG

2(EG – F2) ,

K (u, v) = LN – M2

The mean and Gaussian curvatures are related by the inequality

H2– K = (k1+ k2)2

4 – k1k2 =

(k1– k2)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 =0and K =0at 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+ q2) –2pqs + t(1+ p2)

2(1+ p2+ q2)3 , K =

rt – s2

(1+ p2+ q2)3, where the following notation is used:

p = z x, q = z y, r = z xx, s = z xy, t = z yy, h=

1+ p2+ q2

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 = k1k2>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 – M2 >0 In the special case k1 = k2,

the point is umbilical (circular): R = const for all normal sections at this point.

2 A point M at which K = k1k2<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 – M2 <0

3 A point M at which K = k1k2=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 – M2 =0

Any umbilical point (k1 = k2) is either elliptic or parabolic

of constant... 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... 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 –