Vector Functions, Line Integrals, Directional Derivatives

Question 1

What is the parametric equation for a circular helix

Answer 1

Parametric equation for a circular helix is:

R(t) = acos(t)i + bsin(t)j + ctk

Question 2

When does r(t) tend to r^0 as t tends to t0

Answer 2

r(t) tends to r^0 as t tends to t0 iff:

Norm( r(t)-r^0) tends to 0 as t tends to 0

Question 3

When is r(t) Continuous for all T0

Answer 3

R(t) is continuous for all T0 when:

R(t) tends to r(R0) as t tends to T0

Question 4

What is the tangent vector and what direction does it point

Answer 4

Target vector is r’(t) (=/0) at point P of a curve C

Points in direction of increasing t

Question 5

What is the tangent

Answer 5

The tangent (adjective) is any multiple of r’(t) and noun is the line

Question 6

What is the derivative of R1(t) x R2(t)

Answer 6

Derivative of R1(t) x R2(t) is:

R1’(t) x R2(t) + R1(t) x R2’(t)

Question 7

How to integrate vector parametric functions

Answer 7

To integrate vector parametric functions, integrate each component separately and sum them

Question 8

What is arc length formula

Answer 8

Arc length formula is:
Integral t down to T0 norm(r’(y)) dT =
Integral t down to T0 Root( (dx/dT) ^2 + … + (dz/dT)^2)

Question 9

How can s(t) be calculated (distance travelled between time t and T0)

Answer 9

S(t) can be calculated by:
Integral t down to T0 norm(r’(T) dT
Norm(r’(T)) is speed at time T

Question 10

What is the definition of the unit/principal tangent vector

Answer 10

Definition of unit/principal tangent vector is:
T hat(t) = r’(t)/norm(r’(t))

Question 11

What is the unit tangent vector equal to

Answer 11

Unit tangent vector is equal to:

Dr/ds where s is arc length as a parameter of curve c (mod =1)