Calculus III: Engineering Checkpoint

Choose the mathematical object

Before calculating, identify whether your unknown is a point value, a local rate, accumulated scalar material, oriented work or net flux. The notation should reflect that choice. A gradient predicts change; a Hessian describes second-order behavior; an integral combines contributions over a curve, surface or volume.

Visual guide

VISUAL GUIDETemperature contours give a geometric check

T = 21
T = 23.0
T = 24

Contours of T = 20 + x² + 2y² are ellipses. At (1, 1), the gradient (2, 4) points perpendicular to its contour toward greater temperature. The small displacement (0.1, −0.1) points partly against that gradient, predicting a temperature decrease.

A route through a model

State the region and coordinate convention. Record orientation if relevant. Choose a method, list its regularity assumptions, compute, then inspect units and limiting cases. For density, the result should have units of mass. For velocity flux through area, the result has units of volume per time.

Worked example: a temperature estimate

Suppose $T(x,y)=20+x^2+2y^2$ . At $(1,1)$ , $T=23$ and $\nabla T=(2,4)$ . A displacement $(0.1,-0.1)$ predicts change $-0.2$ , so the linear estimate is $22.8$ . The exact value is $22.83$ ; the quadratic remainder $0.1^2+2(-0.1)^2=0.03$ explains the difference. A directional derivative would instead divide the displacement by its length first.

Worked example: a flow balance

A velocity field $\mathbf v=(x,y,-z)$ occupies a unit cube. Divergence is $1$ , so net outward flux is $1$ . Individually the $x=1,y=1,z=1$ faces contribute $1,1,-1$ ; the opposite faces contribute zero. This is a net source model, not an incompressible flow, because divergence is nonzero.

Practice: justify your choice

Maximize $xy$ subject to $x^2+y^2=2$ .
Find the mass of the unit ball for density $\delta=\sqrt{x^2+y^2+z^2}$ .
Find circulation of $(-y,x,0)$ around the unit circle and explain why divergence is not the relevant derivative.

Show worked solutions

The constraint is compact. Since $(x-y)^2\ge0$ , $2xy\le x^2+y^2=2$ . The maximum is $1$ at $(1,1)$ and $(-1,-1)$ .
Spherical coordinates give $4\pi\int_0^1\rho^3d\rho=\pi$ . The extra $\rho^2$ comes from volume, not density.
Curl is $(0,0,2)$ , so Stokes gives $2\pi$ . Circulation measures tangential work; divergence would measure closed-surface outward flux.

Where to review

An orientation error points back to surface flux. A wrong power of radius suggests a missing Jacobian. A correct calculation on an incorrect region requires revisiting bounds, not more algebra. Repeat the relevant worked example with one changed parameter before moving to differential equations.

Further study

MIT OpenCourseWare: Multivariable Calculus provides a full university course with additional lectures and exercises.