Line and surface integrals · Definition

Vector field in the plane and space

multivariable calculusvector calculus

Before you start you need to know

It models quantities with magnitude and direction that vary from point to point, such as wind velocity or gravitational force.

Definition

A vector field \(\mathbf F\) assigns to each point of a region of the plane or of space a vector; its components \(P,Q,R\) are scalar functions of the coordinates.

\[\mathbf F(x,y,z)=\langle P(x,y,z),\,Q(x,y,z),\,R(x,y,z)\rangle\]

In the plane a field is \(\mathbf F=\langle P,Q\rangle\) and in space \(\mathbf F=\langle P,Q,R\rangle\). Each component is a scalar function of several variables, so what you know about them (domain, continuity, evaluation) applies component by component. Fields are added, subtracted and scaled just like vectors, point by point.

To «see» a field, draw at each point an arrow with its vector. For example, \(\langle x,y\rangle\) points radially outward and its length grows with the distance to the origin; \(\langle -y,x\rangle\) has the same length \(\sqrt{x^2+y^2}\) but is perpendicular to the position vector and turns counterclockwise. Curves that follow the arrows are called field lines or flow lines.

Physical examples abound: fluid velocity, an electric or gravitational field, the force on a particle. When a field is the gradient of a scalar function, \(\mathbf F=\nabla f\), it is called conservative and \(f\) is its potential; this structure will make line integrals depend only on the endpoints.

A typical mistake is to confuse a vector field with a scalar function. A field assigns a vector (several numbers) to each point; a scalar function assigns just one. Another is to evaluate only one component and forget that the field at a point is the whole vector.

Think about it

Is the temperature at each point of a room a vector field?

No: it assigns a number to each point, so it is a scalar field. The air velocity is a vector field.

Describe the field \(\langle 0,1\rangle\) in the plane.

It is constant: at every point it equals the same vector, pointing up.

Exercises 10

  1. Let \(\mathbf F(x,y,z)=\langle xy,\,z,\,x+y\rangle\). Compute \(\mathbf F(1,2,3)\).
  2. How many component functions does a vector field in the plane have?
  3. For \(\mathbf F(x,y)=\langle x,\,y\rangle\), which vector is assigned to the point \((3,-2)\)?
  4. Let \(\mathbf F(x,y)=\langle -y,\,x\rangle\). Find \(\|\mathbf F(3,4)\|\).
  5. The field \(\mathbf F(x,y)=\langle -y,\,x\rangle\) equals \(\langle 0,1\rangle\) at one point. Which?
  6. At which point does the field \(\mathbf F(x,y,z)=\langle x-1,\,y+2,\,z\rangle\) vanish?
  7. Let \(\mathbf F=\langle x,0,z\rangle\) and \(\mathbf G=\langle y,y,-z\rangle\). Compute \((\mathbf F+\mathbf G)(1,2,3)\).
  8. Let \(\mathbf F(x,y,z)=\langle x,y,z\rangle\). Where is \(\|\mathbf F\|=2\)?
  9. Let \(\mathbf H=\langle1,0,1\rangle\) and \(\mathbf K=\langle1,1,-1\rangle\) be the values of two fields at a point. What is \(2\mathbf H-\mathbf K\) at that point?
  10. Let \(\mathbf F(x,y)=\dfrac{\langle -y,\,x\rangle}{x^2+y^2}\). What is \(\|\mathbf F(0,2)\|\)?

References

  • James Stewart, Daniel K. Clegg, Saleem Watson, Calculus: Early Transcendentals, 9th edition, Cengage, 2021. §16.1
  • Joel Hass, Christopher Heil, Przemyslaw Bogacki, Maurice D. Weir, George B. Thomas Jr., Thomas' Calculus, 15th edition, Pearson, 2023. §16.2
  • Jerrold E. Marsden, Anthony Tromba, Vector Calculus, 6th edition, W. H. Freeman, 2012. §1.1