MZB127 Chap.1 Multivariable Functions and Partial Derivatives
Multivariable Functions and Partial Derivatives
A multivariable function links one output to a tuple of inputs, producing surfaces and contour maps rather than a single plane curve. This chapter distinguishes input points, surface points and level curves, then treats first and second partial derivatives as local coordinate-direction information.
The worked method differentiates symbolically before evaluation and checks every rate through signs, units and nearby surface behaviour. The central reading skill is to keep representations separate. A domain is a set of admissible input pairs, a surface point includes the output height, and a contour is a constant-output condition drawn in the input plane.
Coordinate slices then turn a two-variable surface into familiar one-variable curves. This makes the meaning of a partial derivative concrete: one input moves while the other is fixed. Pure second derivatives track how a coordinate slope bends, while mixed derivatives track how one coordinate slope changes as the other input moves.
Smoothness may justify equality of the two mixed orders, but the requested order should still be followed explicitly. Domain restrictions, derivative units and local sign checks remain available even when no graph is supplied. Together these checks prevent an answer from becoming a detached symbolic calculation and prepare the geometric language needed for gradients, chain rules and optimisation.
What this chapter covers
- 01
Multivariable function inputs and domains
- 02
Surface graphs and level curves
- 03
Coordinate slices and first partial derivatives
- 04
Pure and mixed second partial derivatives
- 05
Units and local slope interpretation
Worked example · free
Evaluate two coordinate rates
- 1Hold y fixed and differentiate with respect to x.
- 1Hold x fixed and differentiate with respect to y.
- 1Substitute the point after the symbolic derivatives are complete.
- 1Interpret each sign as a coordinate-direction rate.
Key terms
- Multivariable function
- A rule assigning one output to each allowed tuple of two or more inputs.
- Level curve
- The input points at which a two-variable function has one fixed output value.
- Partial derivative
- A local coordinate rate found while other independent variables are held fixed.
- Mixed derivative
- A second derivative taken successively with respect to two different coordinates.
Multivariable Functions and Partial Derivatives FAQ
What does a contour line represent?
Every point on one contour has the same function output. The curve lives in the input plane, while the corresponding surface points share a common height. Contour spacing also helps compare how rapidly the output changes across different regions.
Why are partial derivatives not ordinary slopes?
A partial derivative specifies which coordinate changes and which coordinates remain fixed. It is the slope of one coordinate slice through the surface, not the rate along every possible path or direction.
When may mixed derivative order be swapped?
For sufficiently smooth functions the mixed partials agree locally, but the equality depends on regularity. Follow the requested order first and inspect non-smooth or piecewise definitions before assuming the two orders match.
How can a partial derivative answer be audited quickly?
Keep the symbolic derivative visible, substitute the point only once, and attach quotient units. Then take a small positive step in the active coordinate while holding the other inputs fixed. The observed sign of change should agree with the derivative, even though the finite change need not equal the local rate exactly.
Exam move
Rebuild the four distinct objects from memory: input tuple, surface point, level curve and coordinate slope. On every derivative, name the active variable and retain units. Use a small positive coordinate step to check the predicted sign. When a stationary point is suggested, remember that zero in one partial derivative is not enough; both first partials and a later classification test are needed.
Begin revision by taking one formula and producing four linked views: state its domain, evaluate a surface point, derive a level curve and compute both first partial derivatives. Explain aloud where each object lives and what is held fixed. Next use a units table with one row for the output, one for each input and one for each derivative. The quotient units should agree with the verbal interpretation of every rate.
Practise symbolic differentiation before substituting coordinates, then reverse the order once to see why early substitution destroys information. For second derivatives, write the differentiation order beside the notation and check whether smoothness is actually available before swapping mixed derivatives.
Use hand-drawn coordinate slices to predict signs at selected points, and test those predictions with a small positive change in just one coordinate. When checking a candidate stationary point, verify both first partials rather than relying on a flat-looking contour. Finish each practice set with an error log divided into object errors, differentiation errors, evaluation errors and interpretation errors.
Rework one example from each category without notes. In an exam response, make the active variable visible, retain the untouched variables, delay numerical substitution and close with a signed, unit-aware sentence about local change. That sequence supplies a readable chain of evidence even when arithmetic becomes complicated.
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