Problem-solving recipes

A step-by-step procedure for every problem type this course teaches — what to do first, what to do next, and how to tell which case you're in.

What are you being asked to do?

Start with what kind of object you're given and what you're asked to produce from it.

Two vectors, need an angle, a projection, or work done by a force — the angle-and-projection recipe. Two vectors, need an area or a vector perpendicular to both — the cross-product recipe instead.

A line in space, from a point and direction or from two points — the line recipe. A plane in space, from three points or a point and normal — the plane recipe. A distance between a point and a line, or a point and a plane — the distance recipe (the two cases use different formulas). An equation in $x,y,z$ you don't recognize — the quadric-surface recipe sorts it by type.

A space curve $\mathbf r(t)$ , need its arc length or a unit-speed reparametrization — the arc-length recipe. The same curve, need curvature and the normal direction — the curvature recipe, which extends into the full TNB frame and torsion once those are also asked for.

A two-variable limit you suspect doesn't exist — the two-path recipe disproves it without ever finding the limit's value.

A surface $z=f(x,y)$ at a point, need a tangent plane or a linear approximation — the tangent-plane recipe, which extends into an error estimate once you're propagating measurement uncertainty instead of just approximating a value. A composite function built from several intermediate variables — the chain-rule recipe. An equation $F(x,y)=0$ you can't solve for $y$ , need $dy/dx$ anyway — the implicit-differentiation recipe. A local approximation better than linear — the second-order Taylor recipe.

A direction and a function of two or more variables, need a rate of change in that direction, or the direction of fastest increase — the directional-derivative recipe.

A critical point, need to classify it — the second-derivative test. An extremum over an entire closed region, not just a single point — the closed-region recipe. An extremum subject to a constraint — Lagrange multipliers, one constraint or two depending on how many the problem gives you.

A double integral over a region that isn't a rectangle — the general-region recipe, which also covers reversing the order of integration. The region or integrand looks circular — convert to polar first. A triple integral, need to pick and set up a coordinate system — the coordinate-choice recipe (rectangular vs. cylindrical vs. spherical). An integral that's hard in the given variables but easier in new ones — change of variables with the Jacobian, for a double or a triple integral.

A vector field, need to know if it's conservative and, if so, find a potential function — the conservative-field-test recipe (2D or 3D, same idea, different verification tool). A line integral of a vector field — evaluate it directly, or shortcut through the potential function if the field is conservative. A circulation or flux integral around a closed curve — Green's Theorem. A flux integral through a curved surface, or its surface area — the flux recipe. A circulation integral over a boundary curve of a surface — Stokes' Theorem, which extends into choosing the easiest surface to bound that curve. A flux integral through a closed surface — the Divergence Theorem, usually the fastest route once the surface enclosed a solid region.

Vectors and the geometry of space

Find the angle between two vectors, a projection, or work done by a force

Use when: you have two vectors, or a force and a displacement, and need the angle between them, a scalar or vector projection of one onto the other, or the work a constant force does

Goal: the angle, projection, or work value the problem specifically asks for, all built from the same dot product and magnitudes

Projections and work

Compute a cross product, then find an area or a normal vector

Use when: you have two vectors in space and need a vector perpendicular to both, the area of the parallelogram they span, or a normal vector to the plane they lie in

Goal: the cross product vector itself, plus whichever of area or normal-vector interpretation the problem asks for

The cross product

Write parametric and symmetric equations of a line in space

Use when: you're given a point and a direction vector, or two points, and need the line through them as parametric or symmetric equations

Goal: parametric equations for x, y, z, and, when no direction component is zero, the matching symmetric equations

Lines in space

Write the equation of a plane in space

Use when: you're given three points on a plane, or a point plus a normal vector, and need the plane's scalar equation ax+by+cz=d

Goal: the coefficients a, b, c, d of the plane's scalar equation

Planes in space

Find the distance from a point to a line or a plane

Use when: you have a point not on a given line or plane and need the shortest distance from it

Goal: a single nonnegative distance value, from whichever of the two formulas matches what the point is being measured against

Distances between points, lines, and planes

Identify and sketch a quadric surface from its equation

Use when: you have an equation in x, y, z with squared terms and need to name which of the six quadric families it belongs to, or recognize it as a cylinder instead

Goal: the surface's family name, plus enough of its cross-sections to sketch it

Cylinders and quadric surfaces

Vector-valued functions and motion

Find the arc length of a space curve and reparametrize by arc length

Use when: you have r(t) and need either its length over an interval, or the same curve rewritten so the parameter itself measures distance traveled

Goal: the arc-length function s(t) and, when it inverts in closed form, the reparametrized curve r(s)

The unit tangent and arc-length parametrization

Find curvature, the unit tangent, and the principal normal

Use when: you have r(t) and need the curvature κ, the unit tangent T, or the principal normal N at a point

Goal: κ, T(t), and N(t), built from r(t)'s first two derivatives

Curvature and the normal vector · extended later in this course

Find the full TNB frame, curvature, and torsion

Use when: you have r(t) and need the complete moving frame T, N, B together with κ and τ, not just T and N alone

Goal: all three frame vectors plus the two turning rates that describe the curve's local shape completely

TNB, torsion, and tangential/normal acceleration

Extends the curvature recipe above once torsion and the binormal are also asked for.

Limits, partial derivatives, and local approximation

Prove a two-variable limit does not exist

Use when: you suspect $\lim_{(x,y)\to(a,b)}f(x,y)$ fails to exist, typically because direct substitution at $(a,b)$ gives an indeterminate form like $0/0$

Goal: two different paths into $(a,b)$ along which $f$ settles on two different numbers, which is already a complete proof the limit does not exist

Paths and nonexistent limits

Find a tangent plane and use it for linear approximation

Use when: you have $z=f(x,y)$ , differentiable at a point $(a,b)$ , and need either the tangent plane's equation or a quick numeric estimate of $f$ at a nearby point

Goal: the tangent plane's equation, and, if a nearby point is given, a linear-approximation estimate of $f$ there

Tangent planes · extended later in this course

Estimate propagated error with the total differential

Use when: $x$ and $y$ are measurements with known uncertainties $dx$ , $dy$ , and you need to know how much that uncertainty propagates into a computed quantity $z=f(x,y)$

Goal: an estimate $dz$ of how far the computed quantity could be off, given the input uncertainties

Linear approximation and differentials

Extends the tangent-plane recipe above for propagating measurement uncertainty rather than approximating a single value.

Apply the multivariable chain rule

Use when: $z=f(x,y)$ and each input is itself a function of one or more other variables, and you need the rate of change of $z$ with respect to one of those underlying variables

Goal: a correct sum of partial-derivative products, one term for every path from $z$ down to the variable being differentiated

The multivariable chain rule

Find dy/dx by implicit differentiation from F(x,y)=0

Use when: an equation relates $x$ and $y$ but solving explicitly for $y$ is hard or impossible, and you need the slope $dy/dx$ at a specific point

Goal: a numeric slope $dy/dx$ at the given point, using only the two partial derivatives of $F$

Implicit differentiation in several variables

Build the second-order Taylor polynomial in two variables

Use when: you need a quadratic, not just linear, local approximation to $f(x,y)$ near a point $(a,b)$ , one that also matches the surface's curvature

Goal: the polynomial $P_2(x,y)$ , matching $f$ 's value, both first partials, and all three second partials at $(a,b)$

Taylor's theorem in two variables

The gradient and directional derivatives

Find a directional derivative and the direction of steepest ascent

Use when: you have $f(x,y)$ and a point, and need either the rate of change in one specific given direction, or the direction and rate that give the fastest possible increase

Goal: a single number $D_{\mathbf u}f$ for a given direction, or a unit vector and its matching maximum rate for steepest ascent

The gradient as the direction of steepest ascent

Optimization

Classify a critical point with the second-derivative test

Use when: you have a critical point $(x_0,y_0)$ of $f(x,y)$ , meaning $f_x=f_y=0$ there, and need to know whether it is a local max, a local min, or a saddle

Goal: a definite classification of the point, or a clear statement that the test can't decide

The second derivative test

Find the absolute extrema of $f$ on a closed, bounded region

Use when: you're asked for the absolute maximum and minimum of $f(x,y)$ over a region $R$ that includes its own boundary, and $R$ is closed and bounded

Goal: the largest and smallest values $f$ actually attains on $R$ , each paired with the point where it happens

Extrema on closed, bounded regions

Solve a constrained optimization problem with Lagrange multipliers

Use when: you need the extrema of $f(x,y)$ subject to a single equation constraint $g(x,y)=k$

Goal: every candidate point for the constrained extremum, with the true max and min picked out by comparing $f$ across all of them

Lagrange multipliers · extended later in this course

Solve a constrained optimization problem with Lagrange multipliers, two constraints

Use when: you need the extrema of $f(x,y,z)$ subject to two equation constraints $g(x,y,z)=k_1$ and $h(x,y,z)=k_2$ at once, with $\nabla g$ and $\nabla h$ never parallel on the intersection

Goal: every candidate point on the curve where both constraint surfaces meet, with the true max and min picked out by comparing $f$ across all of them

Lagrange multipliers with two constraints

Extends the one-constraint recipe above with a second multiplier.

Double and triple integrals

Set up and evaluate a double integral over a general region

Use when: you're asked for $\iint_R f\,dA$ where $R$ 's boundary is one or more curves, not a plain rectangle

Goal: a single number, reached through whichever order of integration the region's bounds make solvable

Reversing the order of integration

Convert and evaluate a double integral in polar coordinates

Use when: $R$ is a disk, sector, or annulus, or the integrand or boundary involves $x^2+y^2$ as a single block

Goal: the double integral's value, computed in $r,\theta$ with no leftover $x$ 's or $y$ 's anywhere

Double integrals in polar coordinates

Choose cylindrical vs. spherical coordinates and set up a triple integral

Use when: you're asked for a triple integral over a solid described by curved surfaces, such as cylinders, cones, paraboloids, or spheres, and rectangular coordinates would need messy square-root bounds

Goal: the right coordinate system for the whole solid, and a triple integral written in it with no leftover square roots

Choosing a coordinate system

Perform a change of variables in a double integral using the Jacobian

Use when: $R$ 's boundary is awkward in $x,y$ (a tilted parallelogram, a region between hyperbolas) but becomes simple under a given or chosen substitution $x=x(u,v),\ y=y(u,v)$

Goal: the integral rewritten over a simpler region $S$ in $u,v$ , with the Jacobian paying for the change

Change of variables in double integrals · extended later in this course

Perform a change of variables in a triple integral using the Jacobian

Use when: $E$ 's boundary is awkward in $x,y,z$ , such as an ellipsoid, but becomes simple under a substitution $x=x(u,v,w),\ y=y(u,v,w),\ z=z(u,v,w)$

Goal: the integral rewritten over a simpler solid $B$ in $u,v,w$ , with the $3\times3$ Jacobian paying for the change

Change of variables in triple integrals

Extends the double-integral change-of-variables recipe above with a third Jacobian row/column.

Vector fields and line integrals

Test a 2D field for conservativeness and find its potential function

Use when: you have a 2D field $\mathbf{F}=(P,Q)$ and need to know whether it is conservative, and if so, produce an actual potential function $f$

Goal: either a confirmed potential function $f$ with $\nabla f=\mathbf{F}$ , or a definitive proof that no such $f$ exists

The 2D test for conservative fields · extended later in this course

Evaluate a line integral of a vector field

Use when: you need $\int_C \mathbf{F}\cdot d\mathbf{r}$ for a specific curve $C$ and field $\mathbf{F}$

Goal: the numerical value of the work integral $\int_C \mathbf{F}\cdot d\mathbf{r}$

The Fundamental Theorem for line integrals

Apply Green's Theorem to a circulation integral

Use when: you need $\oint_C\mathbf{F}\cdot d\mathbf{r}$ around a closed, piecewise-smooth curve $C$ that bounds a region $R$

Goal: the circulation value, computed as a double integral of curl over $R$ instead of a line integral around $C$

Green's theorem · extended later in this course

Surface integrals and flux

Find the surface area of, and flux through, a parametric surface

Use when: you have a parametric surface $\mathbf{r}(u,v)$ over a parameter domain, and need either its surface area or the flux of a field $\mathbf{F}$ through it

Goal: either the surface's true area, or the numerical flux $\iint_S\mathbf{F}\cdot d\mathbf{S}$ , correctly oriented

Flux integrals

The fundamental theorems of vector calculus

Test a 3D field for conservativeness and find its potential function

Use when: you have a 3D field $\mathbf{F}=(P,Q,R)$ on a simply connected domain and need to know whether it is conservative, and if so, produce an actual potential function $f$

Goal: either a confirmed potential function $f$ with $\nabla f=\mathbf{F}$ , or a definitive proof that no such $f$ exists

The 3D conservative-field test

The 3D generalization of the 2D conservative-field test above, using curl instead of a single partial-derivative check.

Apply Green's Theorem, tangential or normal form

Use when: you need either $\oint_C\mathbf{F}\cdot d\mathbf{r}$ (circulation) or $\oint_C\mathbf{F}\cdot\mathbf{n}\,ds$ (flux) around a closed, piecewise-smooth curve $C$ bounding a region $R$

Goal: the circulation or flux value, whichever was asked for, computed as one double integral over $R$

Green's theorem, two forms

Extends the circulation-form recipe above to also cover the flux (normal) form.

Apply Stokes' Theorem to a circulation integral

Use when: you need $\oint_C\mathbf{F}\cdot d\mathbf{r}$ around a closed curve $C$ in space, and a surface $S$ with boundary $C$ is given

Goal: the circulation value, computed as a curl-flux integral over $S$ instead of a line integral around $C$

Stokes' theorem · extended later in this course

Apply Stokes' Theorem, choosing the easiest surface for the boundary

Use when: you need $\oint_C\mathbf{F}\cdot d\mathbf{r}$ and the obvious surface spanning $C$ looks painful to parametrize or integrate over

Goal: the same circulation value, computed over whichever surface sharing $C$ 's boundary makes the curl-flux integral easiest

Using Stokes' theorem to change the surface

Extends Stokes' Theorem above with the freedom to pick whichever bounded surface makes the integral easiest.

Apply the Divergence Theorem to a closed-surface flux integral

Use when: you need $\oiint_S\mathbf{F}\cdot d\mathbf{S}$ for a closed surface $S$ fully enclosing a solid $E$

Goal: the flux value, computed as a triple integral of divergence over $E$ instead of a surface integral over $S$

The divergence theorem