MULTIVARIABLE CALCULUS 08
Vector Fields
A vector field assigns a vector to every point in a region of space.
The main idea
A scalar function assigns one number to each point.
For example,
\[ T(x,y) \]might give the temperature at the point \((x,y)\).
A vector field instead assigns a vector:
So every point has both a magnitude and a direction associated with it.
1. A simple example
Consider
\[ \mathbf{F}(x,y) = \begin{pmatrix} x\\ y \end{pmatrix}. \]At
\[ (1,2), \]the vector is
\[ \mathbf{F}(1,2) = \begin{pmatrix} 1\\ 2 \end{pmatrix}. \]At
\[ (-2,1), \]the vector is
\[ \mathbf{F}(-2,1) = \begin{pmatrix} -2\\ 1 \end{pmatrix}. \]2. How to visualise a vector field
At many points in the plane, draw a small arrow representing the vector assigned to that point.
The direction of the arrow shows the direction of the vector.
Its length can represent the magnitude.
3. Radial vector field
For
\[ \mathbf{F}(x,y) = \begin{pmatrix} x\\ y \end{pmatrix}, \]the vectors point away from the origin.
The farther the point is from the origin, the larger the vector magnitude becomes.
Since
\[ \|\mathbf{F}\| = \sqrt{x^2+y^2}, \]the magnitude equals the distance from the origin.
4. Diagram
5. Rotational vector field
Consider
\[ \mathbf{F}(x,y) = \begin{pmatrix} -y\\ x \end{pmatrix}. \]At
\[ (1,0), \]the vector is
\[ \begin{pmatrix} 0\\ 1 \end{pmatrix}. \]At
\[ (0,1), \]the vector is
\[ \begin{pmatrix} -1\\ 0 \end{pmatrix}. \]The vectors are tangent to circles centred at the origin.
So this field represents rotational motion.
6. Vector fields in physics
A velocity field may describe the velocity of fluid at each point:
\[ \mathbf{v}(x,y,z). \]A gravitational field assigns a force direction and magnitude to each position.
Electric and magnetic fields are also vector fields.
7. Vector fields in differential equations
A system
\[ \frac{dx}{dt}=f(x,y), \] \[ \frac{dy}{dt}=g(x,y) \]can be written as
where
\[ \mathbf{F}(x,y) = \begin{pmatrix} f(x,y)\\ g(x,y) \end{pmatrix}. \]So the right-hand side of a system of ODEs is a vector field.
8. What the field means in an ODE system
At each state \((x,y)\), the vector
\[ \mathbf{F}(x,y) \]gives the instantaneous direction and speed of movement through the phase plane.
9. Trajectories follow the vector field
A solution trajectory
\[ \mathbf{x}(t) \]must satisfy
\[ \mathbf{x}'(t) = \mathbf{F}(\mathbf{x}(t)). \]So the tangent vector to the trajectory always agrees with the vector field.
10. Scalar field versus vector field
A scalar field assigns one number:
\[ f(x,y). \]A vector field assigns several components:
\[ \mathbf{F}(x,y) = \begin{pmatrix} P(x,y)\\ Q(x,y) \end{pmatrix}. \]11. Gradient fields
If a scalar function
\[ \phi(x,y) \]is given, its gradient
\[ \nabla\phi \]is itself a vector field.
For example, if
\[ \phi(x,y)=x^2+y^2, \]then
\[ \nabla\phi = \begin{pmatrix} 2x\\ 2y \end{pmatrix}. \]This is a radial vector field.
12. Conservative vector fields
A vector field of the form
\[ \mathbf{F}=\nabla\phi \]is called a conservative vector field.
The scalar function \(\phi\) is called a potential function.
Such fields have important properties when we study line integrals.
13. Divergence: the idea
A vector field may appear to spread outward from a point or converge inward.
The divergence measures this local source-or-sink behaviour.
For
\[ \mathbf{F} = \begin{pmatrix} P\\ Q \end{pmatrix}, \]the divergence is
14. Example of divergence
For
\[ \mathbf{F}(x,y) = \begin{pmatrix} x\\ y \end{pmatrix}, \]we have
\[ \frac{\partial P}{\partial x}=1 \]and
\[ \frac{\partial Q}{\partial y}=1. \]Therefore,
\[ \boxed{ \nabla\cdot\mathbf{F}=2 } \]The positive divergence agrees with the outward-spreading pattern of the field.
15. Curl: the idea
Another question is whether a vector field tends to rotate locally.
In two dimensions, a useful scalar form of curl is
A nonzero value indicates local rotational tendency.
16. Example of rotational behaviour
For
\[ \mathbf{F}(x,y) = \begin{pmatrix} -y\\ x \end{pmatrix}, \]we have
\[ P=-y, \qquad Q=x. \]Therefore,
\[ \frac{\partial Q}{\partial x}=1 \]and
\[ \frac{\partial P}{\partial y}=-1. \]So
\[ \boxed{ \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 2 } \]which agrees with the rotational pattern.
17. Three-dimensional vector fields
In three dimensions,
\[ \mathbf{F}(x,y,z) = \begin{pmatrix} P(x,y,z)\\ Q(x,y,z)\\ R(x,y,z) \end{pmatrix}. \]The divergence is
and the curl is a vector:
\[ \nabla\times\mathbf{F}. \]18. Why vector fields matter in modelling
Whenever a system has several state variables, the model determines a vector of rates.
For example,
\[ \begin{pmatrix} dS/dt\\ dI/dt\\ dR/dt \end{pmatrix} \]forms a vector field in the state space of an epidemic model.
Every possible state
\[ (S,I,R) \]has an associated direction of motion.
19. The central idea
Vector fields turn multivariable functions into geometric descriptions of flow, force and dynamical change.