Dimensions in Differential and Integral Relations
If y has dimensions [Y] and x has dimensions [X], then [dy/dx]=[Y][X]^-1 and [integral y dx]=[Y][X], provided the derivative or integral is taken with respect to the stated physical variable.
Why this shows up in the exam
Reading graph slopes · Checking differential equations · Interpreting accumulated physical quantities
Learn the idea
Differentiation divides by the independent variable's dimension, while integration multiplies by it. A derivative is a change per change and an integral is accumulated slices, so their dimensional effects follow the denominator or slice width.
🧠 Memory hook: Derivative divides by the axis; integral multiplies by the slice.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- [dy/dx] = [y]/[x] — dimension of a derivative
- [integral y dx] = [y][x] — dimension of an integral
- [dⁿ y/dxⁿ] = [y][x]^-n — dimension after n derivatives
How to approach it
- 1Identify dependent and independent variables
- 2Apply divide or multiply for each operation
- 3Check every term of the resulting equation
Common slip-ups that cost marks
- •Assigning the graph ordinate's dimension to its slope
- •Ignoring repeated derivatives
- •Integrating without including the differential's dimension
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A quantity Q is defined as Q = force^1 / speed^2. Which dimensional formula represents Q?
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