Chapter 6 - Gradient Of A Scalar Field

Question 1

Grad (phi)

Answer 1

= (d(phi)/dx, d (phi)/dy, d (phi)/dz)

Question 2

Derivative of phi at P in the direction u hat

Answer 2

Grad*phi = (phi (p’) - phi (p)) / h

Question 3

Grad*phi

Answer 3

U hat• grax phi =

|grad phi|cos theta where theta is the angle between grad phi and u hat

Question 4

Level set of a map F

Answer 4

Lc (F) = {(x1, x2, … , xn) | F (x1, x2, … , xn) = c}

Question 5

Level surface or isosurface

Answer 5

For a scalar field phi (x,y,z) over R^3, the level set for a given value c represents a surface, the level surface

Question 6

Level curve, contour line or isoline

Answer 6

For a scalar field phi (x,y) over R^2, the level set for a given value c represents a curve, the level curve

Question 7

Theorem?

Answer 7

Let P be any point in the level surface.
Lc (phi) = {(x,y,x) | phi (x,y,z) = c} such that (grad phi)p = 0.
Then (grad phi)p = | (grad phi)p | n hat,
where n hat is a unit vector normal to the level surface at P

Question 8

Integral (from A to B) of [grad phi • dr]

Answer 8

= phi (B) - phi (A)