13.1 Vector Fields
(we’ll start with 2-dimensional vector fields)
Examples of vector fields:
Figure 1a&b – Air velocity vectors that indicate wind speed and direction . Can you tell the where the wind speeds were the fastest? Thicker arrows indicate faster wind speeds (we will be drawing vectors with a longer length instead of thickness). They are also using different colors to indicate different speeds. This is a velocity vector field.
Figure 2 – Ocean currents (a) Air flow past an inclined airfoil (b)
Both velocity vector fields.
Figure 13 – Velocity field in fluid flow – where is the speed of the fluid the fastest? (the longer vectors are inside the smaller radius which indicates their speed is faster)
Figure 14 – Gravitational force field –associates a force vector with each point in space. Newton’s Law of Gravitation.
Example – Electric force fields – electric charge Coulomb’s Law
F i g u r e 1 5 – o u r o l d g r a d i e n t v e c t o r , , x y f f f ∇ = 〈 〉 , “ d e l l f ” i s a g r a d i e n t v e c t o r f i e l d . F i g u r e 1 5 s h o w s a
contour map of the function in red with the gradient vector field in blue. Notice again the gradient vectors are perpendicular to the level curves. Notice the gradient vectors long where the level curves are close together (short where the level curves are far apart). Closely spaced level curves indicate a steep graph and the length of the gradient vector is a value of the directional derivative of the function, so those values will be larger; long gradient vectors. Level curves far apart indicate a less steep slope so the value of the directional derivative will be smaller; short gradient vectors.
It is bold letter in the book, but I want to see → above vectors!
r Definition: A vector field in 2-dimensions is a function F that assigns to each point (x, y) in the domain a r 2-dimensional vector F (x, y).
r *the Domain is a set of points, (x, y) , the Range is a set of vectors, F (x, y) .
r rr Written in terms of its component functions, P and Q, F (x, y) = P(x, y)i + Q(x, y) j or 〈P(x, y), Q(x, y)〉 .
P and Q are scalar fields, not vector fields.