Rates of change of multivariable functions with respect to one variable at a time.
For f(x,y), the partial derivative ∂f/∂x holds other variables fixed. Mixed partials commute for smooth functions. The total differential df=f_x dx+f_y dy estimates changes and propagates measurement errors.
For P=nRT/V, ∂P/∂T = nR/V.
Systems with several variables change in several directions, and partial derivatives isolate the effect of each variable.
Rates of change of multivariable functions with respect to one variable at a time.
Systems with several variables change in several directions, and partial derivatives isolate the effect of each variable.
A partial derivative holds other variables fixed. A total derivative includes all dependencies, so it adds the chain-rule contributions of each variable.
A matrix of all first partial derivatives of a vector function. It maps small input changes to output changes.
Typical applications include finding how stress changes with temperature while load is held constant, computing the sensitivity of a circuit output to each component and estimating the uncertainty in a volume from measured dimensions.
Study "Taylor and Maclaurin series" first (topic 40 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
Chain rule for several variables. Error propagation uses partials. Thermodynamic properties use partials constantly.
For P=nRT/V, ∂P/∂T = nR/V.
The next topic is "Gradient", which builds directly on this one.
It is classed as Intermediate in Engineering Mathematics (topic 41 of 104). It is an intermediate topic. Work through the beginner topics in Engineering Mathematics first.
Want to practise? Open the interactive course for quick checks, quizzes, simulations and progress tracking.