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MA35 Labs

1 Single Variable Calculus

2 » Two Variable Calculus

3 Three Variable Calculus

Polar Coordinate Labs

Two Variable Functions

Contents

2.1 Functions of Two Variables

2.2 Differentiation

2.3 More Derivatives

2.4 Integration

2.5 Vector Analysis

2.5.1 Parametric Curves and Surfaces

2.5.2 Path Integrals

2.5.3 Line Integrals, Circulation, and Flux

2.5.4 Vector Fields, Curl and Divergence

2.5.5 Conservative Vector Fields

2.5.6 Green's Theorem

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Vector Fields, Curl and Divergence

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A vector field is defined by a function which associates a vector with each point in the domain of the function.

In two dimensions, a vector field V can be expressed in terms of the functions p(x,y) and q(x,y) as V(x,y) = (p(x,y),q(x,y)).

Note that for the gradient vector field, p = f_x and q = f_y.

The divergence of a vector field V(x,y) = (p(x,y),q(x,y)) is defined as div V = ∇ ⋅ V = p_x + q_y. Note that divergence of a vector field is a scalar value.

The scalar curl of a two-dimensional vector field is defined as scalar curl V = -p_y(x,y)+q_x(x,y). The curl of a vector field V is usually defined for a vector field in three variables by the condition curl V = ∇ x V. If the third coordinate is 0, then curl(p(x,y),q(x,y),0) = ∇ {times} (p(x,y),q(x,y),0) = (0,0,q_x-p_y). The third coordinate, -p_y(x,y)+q_x(x,y) is called the scalar curl of V. The scalar curl of a vector field in the plane is a function of x and y and it is often useful to consider the function graph of the (x,y,-p_y(x,y) + q_x(x,y)).

If a two-dimensional vector field F(p,q) is conservative, then its curl is identically zero since p_y = q_x, ∇ {times} F = (0,0,q_x-p_y) = (0,0,0).

Demos

Divergence and Curl in 2D
	This demo shows, in the first window, the vector field V(x,y). In the "Divergence" window, the z-coordinate of the red surface indicates the value of the divergence at (x,y): div((x,y)) = f(x,y) = z. In the "Curl" window, the z-coordinate of the green surface indicates the value of the scalar curl at (x,y).