Vector calculus.
School-level vector calculus through parametric curves, Bézier paths, motion, tangents and arc length.
Work through the topic
Learn the method, follow the examples, then try the worksheet.
- 01
Parametric descriptions of curves
Describe curves using coordinates with a common parameter, retaining direction and restrictions.
- 02
Coordinates varying at constant rates
Model constant velocity and distinguish a collision from a geometric path intersection.
- 03
Polynomially varying coordinates
Analyse polynomial coordinate paths, including reversals and coordinate turning points.
- 04
Bézier paths
Construct Bézier paths and use control points to find endpoint tangent directions.
- 05
Vector-valued position functions
Represent position in two or three dimensions and calculate displacement between parameter values.
- 06
Differentiating and integrating components
Differentiate and integrate vector functions component by component, preserving vector constants.
- 07
Velocity, acceleration and speed
Calculate velocity, acceleration and speed and distinguish a change of direction from a change of speed.
- 08
Parametric gradients and tangent directions
Find tangent gradients and direction vectors for parameterised curves, including vertical tangents.
- 09
Sine/cosine parameterisations and circular motion
Model circles, ellipses and helices using trigonometric coordinates and interpret period and direction.
- 10
Arc length in two and three dimensions
Calculate distance along a parameterised path by integrating speed in two or three dimensions.
Topic review
Revisit the key ideas, then choose a practice paper.