Definitions

Question 1

linearly dependent

Answer 1

The vectors {x1, x2, . . . , xm} in vector space V are linearly dependent if there exists a non-trivial linear combination of these vectors
such that it is equal to zero

Question 2

proposition of linear dependence

Answer 2

Vectors {x1, x2, . . . , xm} in vector space V are linearly
dependent if and only if at least one of these vectors is expressed via linear
combination of other vectors:

Question 3

define basis

Answer 3

Let V be n-dimensional vector space. The ordered set {e1, e2, . . . , en} of n linearly independent vectors in V is called a basis of the vector space V .

Question 4

define scaler product

Answer 4

Scalar product in a vector space V is a function B(x, y)
on a pair of vectors which takes real values and satisfies the the following conditions:
(symmetricity condition)
(linearity condition)
(positive-definiteness condition)

Question 5

define dimension

Answer 5

Vector space V has a dimension n if there exist n linearly independent vectors in the vector space

Question 6

define orthonormal basis

Answer 6

ei such that they satisfy δij

Question 7

explain what is meant by two bases having the same orientation?

Answer 7

We say that two bases {e1, . . . en} and {e1, . . . en} in V have
the same orientation if the determinant of transition matrix (1.39) from the first basis to the second one is positive: det T > 0.

Question 8

Euler theorem ?

Answer 8

P = orthogonal and preserves orientation. then it is a rotation w.r.t θ and axis l. 
P(N) = N 
trP = 1 +2cos(θ)

Question 9

define a differential 1-form

Answer 9

is a function on tangent vectors of E^n such that it is linear at each point:
ω(r, λv1 + μv2) = λω(r, v1) + μω(r, v2)

Question 10

area of a parallelogram ?

Answer 10

S(x,y) = S(Π(x,y)) = |x×y| = |x|•|y||sinθ|

Question 11

volume of a parallelpiped ?

Answer 11

V({a,b,c}) = |a•(b×c)|

Question 12

velocity vector?

Answer 12

v(t) = dr/dt

Question 13

define equivalent curve?

Answer 13

r1(t):(a,b)→E^n & r2(t):(c,d)→E^n are equivalent if there exists reparameterisation map t(τ):(a,b)→(c,d) such that r2(τ) = r1(t(τ))

Question 14

define orientation of curves

Answer 14

two curves have opposite orientation if their reparameterisation t(τ) has a negative t’(τ)

Question 15

ω?

Answer 15

? df = ∂f/∂x.dx + ∂f/∂y.dy

Question 16

df(A)?

Answer 16

df(A) = ∂_A.f =Ax.∂f/∂x + Ay.∂f/∂y

Question 17

differentiable 1-form is exact?

Answer 17

if there exists a functoion f such that ω = df

Question 18

∫c ω ?

Answer 18

∫ω(v(t))dt = ∫[a.dx/dt + b.dy.dt ]dt

Question 19

if ω is exact. ∫c ω ?

Answer 19

∫c ω = f(r(t_1)) - f(r(t_2))

Question 20

length of a curve?

Answer 20

L = ∫|v(t)|dt

Question 21

define natural parameter ?

Answer 21

is s=s(t) on the curve r=r(t) is a parameter which define the length of the arc of the curve between initial point r(t1) and the point r(t).

Question 22

define curvature ?

Answer 22

curvature of the curve in a given point is equal to the modulus (length) of acceleration vector (normal acceleration) in natural parameterisation