PHYS2150 Chap.2 Partial Derivatives and Local Approximation
Partial Derivatives and Local Approximation
What the chapter connects
Limits establish whether a function settles to one value, differentiability supplies the local plane, the gradient selects a direction, and Taylor expansion extends that plane with curvature.
Treating these as one chain prevents a formula chosen for one question from being used to answer another.
Local change needs a direction
A multivariable function can change differently along different approaches, so a derivative statement is incomplete until its direction is named. A partial derivative varies one coordinate while holding the others fixed, matching the slope of a cross-section through the graph.
The gradient collects the first partial derivatives into a vector. At a point, it identifies the direction of greatest increase, its magnitude gives that maximum rate per unit distance, and it is normal to the level curve or level surface through the point.
Directional derivatives therefore require a unit direction vector before the dot product is taken.
Limits separate refutation from proof
Path tests are powerful because two different path values immediately prove that a multivariable limit does not exist. Agreement along several straight lines does not prove existence, since curved approaches remain. When basic paths agree, switch from examples to a bound.
Polar coordinates can isolate a radial factor that tends to zero while the angular factor remains bounded, producing one argument that covers every direction.
The logical roles must stay separate: paths search for contradictions; inequalities, polar bounds or a direct epsilon argument establish a common limiting value.
Differentiability means one plane controls the error
Existing partial derivatives do not by themselves guarantee differentiability.
Differentiability requires a single linear expression to approximate the total increment, with a remainder that becomes negligible relative to the displacement. Continuous first partial derivatives in a neighbourhood give a standard sufficient condition, while isolated exceptional points need direct analysis.
Once differentiability is established, the tangent plane and local linearisation come from the function value and gradient at the base point. Every derivative is evaluated at that point, while the displacement connects the base point to the nearby target.
Dependency paths organise the chain rule
The multivariable chain rule becomes reliable when dependence is drawn before differentiation.
If an output depends on x and y and both depend on time, each directed path from time to the output contributes a product of derivatives. With two intermediate variables depending on two parameters, the same path rule produces one partial derivative for each parameter. Units supply an independent check: every product contributing to a total derivative must reduce to the same output-per-input unit.
A missing path often appears as a missing physical influence or an incompatible unit.
Taylor terms measure local curvature
A first-order Taylor approximation is the tangent-plane model. A quadratic approximation adds the pure second derivatives and the mixed derivative, showing how curvature changes the linear prediction.
The expansion point and increments must be stated because the same polynomial coefficients do not describe every neighbourhood. Label terms by degree before substitution, retain the mixed term, and distinguish an approximation from an equality.
The remainder describes the omitted derivative order somewhere between the base and target points; without a derivative bound, smaller displacement supports greater confidence but does not create an exact error guarantee.
What this chapter covers
- 01
Multivariable limits and paths
- 02
Partial derivatives and gradients
- 03
Differentiability and tangent planes
- 04
Multivariable chain rules
- 05
Taylor approximation and remainder
- 06
Directional derivatives and level sets
Worked example · free
A linear estimate
- 2Evaluate the base value.
- 2Dot the gradient with the displacement.
- 2State neglected order.
Key terms
- Directional Derivative
- The rate of change per unit distance in a specified unit direction.
- Differentiability
- The existence of a linear local approximation whose error is small relative to displacement.
- Gradient
- The vector of first partial derivatives that points toward greatest local increase.
- Tangent Plane
- The plane supplied by the first-order local model of a differentiable surface.
- Level Set
- The points at which a multivariable function has one fixed value.
- Taylor Remainder
- The part of a local expansion left after the chosen derivative order is retained.
Partial Derivatives and Local Approximation FAQ
Do existing partial derivatives prove differentiability?
No. Partial derivatives inspect coordinate directions, while differentiability requires one linear approximation to control simultaneous changes in all directions. Continuous first partial derivatives nearby give a standard sufficient condition, but exceptional points need direct analysis of the total increment and its remainder.
Can several paths prove a multivariable limit?
Several agreeing paths provide evidence but not proof because infinitely many approaches remain, including curved paths. Two paths with different values disprove existence immediately. When searches do not find a contradiction, a proof usually uses an absolute-value bound, polar coordinates or a direct epsilon argument.
Why normalise a direction vector?
A directional derivative measures change per unit distance. Dotting the gradient with a vector of arbitrary length scales the result by that length and no longer reports the requested unit-direction rate. Divide by the vector magnitude first, then check the resulting rate against the gradient magnitude.
When is a quadratic Taylor approximation worth using?
Use it when the requested accuracy or displacement makes curvature relevant. Evaluate all first and second derivatives at one base point, include the mixed term, and label both increments. Compare with the linear estimate, but do not treat agreement as a rigorous remainder bound without derivative information.
Exam move
Separate revision into four decision routines. For a limit, try two structurally different paths, then move to a bound if no contradiction appears. For differentiability, write the proposed linear part and identify what must happen to the remainder relative to displacement.
For a chain rule, draw every dependency arrow and translate each complete path into one derivative product; use units to check that all products can be added. For a local estimate, write the base point, displacement and derivative order before calculating. Compare first- and second-order approximations when curvature may matter, but do not call their agreement a formal error bound.
Keep a correction log that distinguishes logical errors from algebraic ones: agreeing paths treated as proof, an unnormalised direction, derivatives evaluated at the target instead of the base point, a missing dependency path, or a missing mixed Taylor term. Rework each logged error with new numbers until the setup can be produced without prompts.
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