Vector Functions Etc 2

Question 1

What is the definition of principal normal vector

Answer 1

Definition of principal normal vector is:

N hat = T hat’ / norm(T hat’)

Question 2

What is the curvature and its radius

Answer 2

Curvature is k where k = norm(dT hat/ds) and 1/k is radius of curvature (largest circle that fits locally into curve)

Question 3

What does k equal

Answer 3

K equals norm(T hat’(t))/ norm(r’(t)) comes from the fact k = norm(d^2r/ds^2) then chain rule

Question 4

What is the definition of a line integral I

Answer 4

Definition of a line integral I is:

I = integral b down to a F(r(t)) .r’(t) dt where F(r(t)) = F(x(t),…,z(t))

Question 5

What is the definition of a scalar line integral p(r)

Answer 5

Definition of a scalar line integral of scalar function p(r) is:
I = integral p(r) ds = integral b down to a p(r(t)) * norm(r’(t)) dt

Question 6

What is the definition of a level curve and how can we define a function g on this curve

Answer 6

Definition of a level curve is:
Level curve at c is defined by all points that satisfy f(x,y) = c
In 3D points ( x,y,z) define contour of f at level c
We can define a function g on this curve by :
G(t) = f(x(t),y(t))

Question 7

What is the definition of directional derivative of a function f along a curve C

Answer 7

Definition of directional derivative of a function f along a curve C is:
Dg/dt = Pdf/Pdx *dx/dt + Pdf/Pdy * dy/dt = fx dx/dt + fy dy/dt

Question 8

What is the directional derivative of a function f(x,y) at a point (a,b) and in direction of vector u is given by

Answer 8

Directional derivative of a function f(x,y) at a point (a,b) and in direction of vector u = (u,v) is given by:
Duf = ufx + vfy

Question 9

What is the vector delta/grad f(x,y)

Answer 9

Vector delta/grad f(x,y) is:
Delta f(x,y) = fx(x,y) i + fy(x,y) j

Question 10

What is the grad symbol

Answer 10

The grad symbol is a vector differential operator t which acts on a scalar function and is given by:
Grad = i Pd/Pdx + j Pd/Pdy

Question 11

What is Duf equivalent to (f(x,y) indirection of u)

Answer 11

Duf is equivalent to vector u.grad(f)