Week 4 Scalar & Vector Fields and the Grad operator

Question 1

what is a field

Answer 1

entity whose values depend on position

Question 2

what is the difference between a scalar and vector field

Answer 2

for a scalar field on magnitude depends on position however for a vector field both magnitude and direction depend on position

Question 3

give an example of a scalar and a vector field

Answer 3

scalar - pressure field

vector - electric field

Question 4

what is true of field lines

Answer 4

they are tangential to the field

Question 5

what is the general grad operator equation

Answer 5

sum of the partial derivatives

Question 6

what happens when we apply a grad operator on a scalar field

Answer 6

we get a vector field

Question 7

what are the four properties of the grad operator

Answer 7

1 - ∇f is a vector field
2 - ∇f defines the max rate of change of f
3 - Direction of ∇f is perpendicular to the contours of constant f
4 - the unit vector normal to a normal surface is defined by ∇f / |∇f|

Question 8

what are the three practical uses of ∇f

Answer 8

1 define the direction of max rate of change of a scalar field
2 calculating rate of change of a scalar field in a given direction
3 calculating the unit-vector normal to a level surface

Question 9

what is the procedure to find the unit vector @(x,y,z) normal to the level surface

Answer 9

1 determine ∇f as a function of x,y,z
2 evaluate @ the given point x,y,z to get vector n
3 divide the vector n by |n| to get the unit vector

Question 10

what is the procedure to find the rate of increase of the field, f, @(x,y,z) in the direction between two points

Answer 10

1 determine ∇f evaluated at x,y,z
2 determine ds = (sum of change in variables)
3 determine u = ds/|ds|
4 df/ds = ∇f * u = directional derivative