Vector Fields
Assigning a direction and magnitude to every point in space.
Introduction
A Vector Field is a function that takes a point or and returns a vector.
Common examples include wind velocity maps, ocean currents, and gravitational force fields.
Gradient Fields
Conservative Fields
If is a scalar function (potential function), its gradient is a vector field:
These special fields are called Conservative Vector Fields. The vectors point "uphill".
Visualizing Fields
Interactive: Rotational Field
The red vectors show the direction of flow. In a rotational field, they are tangent to circles.
Worked Examples
Example 1: Sketching a Field
Describe the field .
1. Check Points:
- At (1,0): (Points Right).
- At (0,1): (Points Up).
- At (-1,-1): (Points Down-Left).
2. Pattern:
At every point, the vector points away from the origin.
Length increases with distance.
Conclusion: Radial Explosion Field.
Example 2: Gradient Field
Find the gradient vector field of .
1. Partial Derivatives:
.
.
2. Vectorize:
.
This field is perpendicular to the level curves of .
Example 3: Inverse Square Field
Write the formula for a gravitational field pointing to origin with magnitude inversely proportional to square of distance.
1. Direction:
Unit vector towards origin: .
2. Magnitude:
.
3. Combine:
.
Result: .
Practice Quiz
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