Michaelmas

Question 1

curvature k at a point of the curve is independent of the

Answer 1

parametrisation

Question 2

Four vertex theorem

Answer 2

always at least four vertices on a smooth regular simple closed plane curve

Question 3

principal curvatures

Answer 3

maximum and minimum if curvatures at a point of a surface

Question 4

product of 2 principal curvatures at a point is the

Answer 4

Gaussian curvature

Question 5

a is a smooth curve if all component functions are

Answer 5

smooth maps

Question 6

the image of the interval under a curve is called

Answer 6

the trace of the curve

Question 7

regular parametrised curve

Answer 7

derviative is not 0

Question 8

unit speed curve

Answer 8

vector length is 1 everywhere. also called arc length parametrised curve

Question 9

a parameter change for a smooth regular curve is

Answer 9

a map such that h is smooth, derivative of h is not 0, h(J) = I

Question 10

reparametrisation is orientation preserving if

Answer 10

h’ > 0 and orient reversing if h’ < 0

Question 11

length of smooth regular curve =

Answer 11

length a reparametrisation of curve

Question 12

unit normal vector of a smooth regular plane curve is obtained by

Answer 12

anti-clockwise rotation of the unit tangent vector of the curve at pi / 2

Question 13

the vector t’(s) of a smooth unit speed plane curve is parallel to

Answer 13

vector n(s)

Question 14

signed curvature of a unit speed plane curve is

Answer 14

t’(s) = k(s)*n(s)

Question 15

if a curve turns left curvature is

Answer 15

positive. if turns right its curvature is -ve.

Question 16

point, alpha(u(0)), is an inflection point of smooth regular plane curve with curvature k if ………. and vertex if ….

Answer 16

k(u(0)) = 0 ,k’(u(0)) = 0

Question 17

isoperimetric inequality: (length of smooth regular simple closed plane curve)^2 >=

Answer 17

4piarea of domain enclosed by curve. equal only if curve describes a circle

Question 18

Fundamental theorem of local theory of plane curves

Answer 18

given an open interval and a smooth function, s(0), there exists a unique smooth unit speed plane curve, alpha, with curvature k where alpha(s(0)) = a and alpha’(s(0)) = v(0)

Question 19

evolute is singular iff

Answer 19

curve is a vertex

Question 20

if evolute is regular then its tangent vector is parallel to

Answer 20

the unit normal vector

Question 21

if b is involute of a then

Answer 21

ap is evolute of curve b

Question 22

vector product is

Answer 22

orthogonal to each vector, antisymmetric and will form a +vely oriented orthonormal basis if vectors are orthogonal

Question 23

if curvature is not 0 the derivaitve of the binormal vector of curve is parallel to

Answer 23

the principal normal vector

Question 24

curvature measures the rate the curve is

Answer 24

bending away from its straights tangent line

Question 25

the plane through the curve spanned by its tangent and normal vectors is the

Answer 25

osculating plane of the curve

Question 26

binormal vector is a unit normal the the osculating plane of a curve so derivative of binormal vector measures

Answer 26

the rate of change of the osculating

Question 27

torsion measures the rate at which the curve is

Answer 27

twisting away from its osculating plane

Question 28

if a smooth regular space curve with nowhere vanishing curvature and an affine plane exists a plane through the origin then

Answer 28

torsion - 0

Question 29

Fundamental theorem of local theory of space curves: for any pair of smooth functions there exists a smooth unit speed space curve where the curve is unique up to ………….. so if another curve with same curvature and torsion that curve =

Answer 29

orientation preserving rigid motions.(x*rotation matrix + translation row vector) on curve

Question 30

a subset is a regular surface for every point there exists an open set and a map if

Answer 30

the map is smooth and a homeomorphism aand partial derivatives are linearly independent

Question 31

linearly independent can not be

Answer 31

a linear combination of each other

Question 32

homeomorphism

Answer 32

bijective, continuos and has a continuous inverse

Question 33

graph of a function of a surface, graphs(smooth function)

Answer 33

graph = (u, v, smooth function(u, v)) and is a regular surface

Question 34

U is an open set and f a smooth function andc a regular value. then the preimage is a

Answer 34

regular surface

Question 35

smooth map is diffeomorphisn if it is

Answer 35

bijective and inverse is smooth

Question 36

surface of revolution is obtained by

Answer 36

rotating a curve in the x-z coordinate plane around the z-axis

Question 37

curves obtained by rotating curve by a fixed angle are

Answer 37

meridans

Question 38

tangent vector to a regular surface at a point on surface is a

Answer 38

tangent vector curve’(0) of a smooth curve

Question 39

U is open, f is a smooth map and c a regular value of f. p is pre-image of c. the tangent plane T(p)S is the Euclidean plane which is

Answer 39

orthogonal to gradf(p)

Question 40

canonical inner product has properties

Answer 40

bilinearity, symmetry and positivity

Question 41

map is smooth at point of regular surface,S, if there exists a

Answer 41

local parametrisation, x:U -> S, with point in x(q) and q in U such that compostion f o x is smooth at q

Question 42

Gauss map of the surface S is a smooth map which assigns to every point

Answer 42

a unit normal vector that is orthogonal to T(p)S

Question 43

a surface is non-orietnable if there exists

Answer 43

no global Gauss map - no way to define the map globally continuously on surface.

Question 44

if surface admits a global Gauss map the surface is

Answer 44

orientable

Question 45

if f is a local isometry and a diffeomorphism then f is

Answer 45

an isometry and surfaces are isometric

Question 46

conformal diffeomorphism

Answer 46

both a conformal map and a diffeomorphism

Question 47

f a conformal map preserves

Answer 47

anlges between tangent vectors

Question 48

Smooth function is

Answer 48

Differentiable everywhere