MTH2010 Chap.9 Change of Variables, Vector Fields and Line Integrals
Change of Variables, Vector Fields and Line Integrals
Week 9 opens the vector calculus part of the unit with three topics listed on the weekly page: change of variables in multiple integrals, vector fields, and integrals along curves. A change of variables replaces x and y by functions of new variables u and v.
A small rectangle in the u-v plane maps to a small parallelogram whose area is the absolute value of the Jacobian determinant times du dv, so the integral picks up that factor. Polar coordinates are the first example, with Jacobian r, which explains the area element used in Week 7. Good substitutions are suggested by combinations of x and y that appear in both the boundary and the integrand.
A vector field attaches a vector to each point, as a velocity or a force does, and is pictured as a grid of arrows. Two kinds of integral along a curve follow. The scalar line integral weights a function by arc length, using ds = |r'(t)| dt, and with the function equal to 1 it gives the length of the curve.
The vector line integral adds up the tangential component of a field and represents work or circulation; it is computed as the integral of F(r(t)) dotted with r'(t), it changes sign when the direction of travel is reversed, and it does not depend on the speed of the parametrisation. Curves made of several pieces are handled one piece at a time.
The source materials for this week list the matching textbook sections in Multivariable Calculus and Vector Calculus.
What this chapter covers
- 01
Jacobian determinant: the local area scale factor of a substitution
- 02
Polar coordinates as a change of variables with Jacobian r
- 03
Choosing substitutions from the shape of the region
- 04
Vector fields: pictures, flow and sign predictions
- 05
Scalar line integrals: ds = |r'(t)| dt and arc length
- 06
Vector line integrals: F(r(t)) dotted with r'(t)
- 07
Orientation: reversing the curve reverses the work
- 08
Piecewise paths and the P dx + Q dy notation
Worked example · free
Arc length of a helix
- 1Velocity: r'(t) = (-3 sin t, 3 cos t, 4).
- 1Speed: |r'(t)| = sqrt(9 sin² t + 9 cos² t + 16) = sqrt(25) = 5.
- 1Length: the integral of 5 from 0 to 2 π is 10 π.
Key terms
- Change of variables
- A substitution expressing x and y in terms of new variables, used to simplify a region or integrand.
- Jacobian determinant
- The determinant of the matrix of partial derivatives of x and y with respect to u and v.
- Vector field
- A rule assigning a vector to every point of a region in the plane or in space.
- Arc length
- The length of a curve, computed as the integral of its speed over the parameter interval.
- Line integral
- An integral of a scalar function or of a vector field's tangential component along a curve.
- Circulation
- The line integral of a vector field around a closed curve.
Change of Variables, Vector Fields and Line Integrals FAQ
What does the Jacobian measure in a change of variables?
Its absolute value is the factor by which small areas are stretched when moving from the u-v plane to the x-y plane, so it multiplies the integrand after the substitution.
Does the direction of a curve affect a line integral?
For a vector field, yes: reversing the direction reverses the sign of the integral. For a scalar line integral with respect to arc length, such as length or the mass of a wire, the direction does not matter.
Does the parametrisation of a curve change a line integral?
Not as long as it traces the same curve in the same direction. Faster or slower parametrisations change r'(t), but the change is exactly cancelled by the change in dt.
How do I compute a line integral along a path with corners?
Split the path into smooth pieces, give each piece its own parametrisation, interval and direction of travel, compute each integral separately, and add the results. A sketch with arrows on every piece prevents sign errors.
Exam move
Treat the Jacobian as a generalised version of the polar r: compute it for polar coordinates once by hand so that the connection is clear, then practise one linear substitution where the new region is a square. For line integrals, rehearse the four moves on every problem: parametrise with the right orientation, substitute into the field, dot with the velocity, integrate.
Keep scalar and vector line integrals in separate columns of your notes, with ds in one and dr in the other, and check signs with a quick sketch of the field along the curve.
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