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Calculus 3

Vectors, partial derivatives, multiple integrals and vector calculus. 3143 problems, in the order most textbooks teach them.

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Six core problems from across the course. Pick one, try it on paper, then open the solution.

Let \( \displaystyle \mathbf{u} = \langle 1, 3, 0 \rangle \) and \( \displaystyle \mathbf{v} = \langle 3, -4, -5 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.

Problem 9.1easy✓ Nihil obstat

Find \( \displaystyle \mathbf{u} \times \mathbf{v} \) for \( \displaystyle \mathbf{u} = \langle 3, 4, -3 \rangle,\ \mathbf{v} = \langle -1, -3, 2 \rangle \), and the area of the parallelogram they span.

Problem 9.18easy✓ Nihil obstat

For \( \displaystyle f(x, y) = \ln{\left(x^{2} + y^{2} + 1 \right)} \), find \( \displaystyle f_x \), \( \displaystyle f_y \) and \( \displaystyle f_{xy} \).

Problem 10.1hard✓ Nihil obstat

Use Lagrange multipliers to find the maximum and minimum of \( \displaystyle f(x, y) = 4 x + y \) on the circle \( \displaystyle x^2 + y^2 = 25 \).

Problem 10.55medium Video✓ Nihil obstat

Evaluate \( \displaystyle \iint_D x^{2} \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).

Problem 11.16easy✓ Nihil obstat

Find the work done by \( \displaystyle \mathbf{F} = \langle - y, x \rangle \) along the unit circle, counterclockwise.

Problem 12.2medium✓ Nihil obstat

Final exam review →

Interactives

Line Integral Explorer

A scroll-through tutorial: straight and curved paths, flow vs flux, work, a whirlpool and a river, with ∫ F · dr worked out as you drag.

Line integrals and work Assumptions stated

Gradient Explorer

A scroll-through tutorial on a contour map: turn to face any direction, read the slope, and see why the gradient points uphill, crosses contours at right angles, and vanishes at a saddle.

Gradient and directional derivatives Assumptions stated

Vectors and space

Dot product and angles

u · v, lengths, and the angle between two vectors.

Cross product

u × v by the determinant, and the area of the parallelogram it spans.

Equation of a plane

A plane through three points, from a normal vector.

Vectors in the plane

Component form, length, unit vectors, combinations, and the direction angle of plane vectors.

Vectors in three dimensions

Vectors in three dimensions: between points, combinations, length, and a vector of given length along a direction.

Projections and orthogonality

Projections, scalar components, and splitting a vector into parallel and orthogonal parts.

Triple products and volumes

The scalar triple product u·(v × w): volumes of parallelepipeds, and testing for coplanar vectors.

Lines in space

Parametric and symmetric equations of lines, and whether two lines are parallel, intersecting or skew.

Cylindrical and spherical coordinates

Converting points between rectangular, cylindrical and spherical coordinates.

Calculus of vector-valued functions

Derivatives, tangent vectors, unit tangents and integrals of vector-valued functions.

Velocity, acceleration and speed in space

Velocity, acceleration and speed of a moving particle, and position from acceleration.

Equations of spheres

Spheres: the standard equation from a center and radius, and the center and radius from a general equation.

Planes: distances and angles

Planes: through a point with a given normal, the distance from a point, and the angle between two planes.

Quadric surfaces in standard form

Rewrite a quadric surface in standard form and name it.

Arc length and curvature of space curves

Curvature of a plane or space curve at a point, κ = ‖r′ × r″‖ / ‖r′‖³.

Projectile motion

Projectiles: time of flight, maximum height and range from a launch speed, angle and height.

Partial derivatives

Partial derivatives

Differentiate in one variable while holding the others fixed.

Gradient and directional derivatives

∇f, and the rate of change in any direction: D_u f = ∇f · u.

Tangent planes

The plane that touches a surface z = f(x, y) at a point.

Local extrema and saddle points

Critical points of f(x, y), classified with the second derivative test.

Lagrange multipliers

Maximize or minimize subject to a constraint: ∇f = λ∇g.

The multivariable chain rule

dz/dt and ∂z/∂s when the variables themselves depend on other variables.

Implicit differentiation with partial derivatives

dy/dx and ∂z/∂x for implicitly defined functions: −F_x/F_y and −F_x/F_z.

Linear approximation in two variables

The tangent-plane approximation L(x, y) = f(a, b) + f_x(x − a) + f_y(y − b), and using it to estimate.

Maximum rate of change

The gradient points uphill: the maximum rate of change is ‖∇f‖, and D_u f = ∇f · u.

Absolute extrema on a closed region

Absolute maximum and minimum on a closed rectangle: interior critical points, edges and corners.

Level curves and domains

Level curves of a function of two variables: their equations, and the one through a given point.

Limits of functions of two variables

Limits of functions of two variables: by substitution, by cancelling, or showing two paths disagree.

Normal lines and normal vectors to surfaces

The normal vector and normal line to a surface F(x, y, z) = c at a point.

Higher-order partial derivatives

Second partial derivatives, Clairaut's theorem, and whether a function is harmonic.

Multiple integrals

Double integrals over rectangles

Iterated integrals and Fubini's theorem.

Double integrals over general regions

Type I and Type II regions: set the bounds, then integrate.

Double integrals in polar coordinates

Disks and circles: dA = r dr dθ.

Triple integrals

Integrate over a box, one variable at a time.

Changing the order of integration

Integrals you cannot do in the given order: sketch the region and swap dx and dy.

Triple integrals over general regions

Triple integrals over regions bounded by planes and simple surfaces, one variable at a time.

Triple integrals in cylindrical coordinates

Triple integrals in cylindrical coordinates: dV = r dz dr dθ.

Triple integrals in spherical coordinates

Triple integrals in spherical coordinates: dV = ρ² sin φ dρ dφ dθ.

Mass and center of mass of a lamina

Mass and center of mass of a thin plate with variable density, by double integrals.

Jacobians and change of variables

The Jacobian ∂(x, y)/∂(u, v) of a change of variables, and the area scaling it measures.

Riemann sums for double integrals

Estimate a double integral with a midpoint (or corner) Riemann sum on a grid of subrectangles.

Average value over a region

The average value of a function of two variables over a rectangle.

Surface area by double integrals

Surface area of z = f(x, y) over a region, by ∬ √(1 + f_x² + f_y²) dA.

Vector calculus

Line integrals and work

∫ F · dr along a path: parametrize, dot, integrate.

Green's theorem

Trade a line integral around a closed curve for a double integral inside it.

Divergence and curl

∇ · F and ∇ × F, component by component.

Conservative fields and potential functions

Test whether a field is conservative, find a potential function, and evaluate a line integral by the fundamental theorem.

Area with Green's theorem

Green's theorem: a circulation around a closed curve as a double integral of ∂Q/∂x − ∂P/∂y, and areas from line integrals.

The divergence theorem

Flux out of a closed surface as the triple integral of the divergence.

Line integrals of scalar functions

Line integrals of a scalar function with respect to arc length, ∫_C f ds.

Flux across a plane curve

Flux of a vector field across a plane curve, ∫_C F · n ds = ∫ P dy − Q dx.

Divergence and curl at a point

Divergence and curl of a vector field at a given point.

Surface integrals

Surface integrals of scalar functions over planes, spheres and cylinders, ∬_S f dS.

Stokes' theorem

Stokes' theorem: a circulation around a closed curve equals the flux of the curl through a surface it bounds.

Differential equations

Second-order linear equations

Constant coefficients: the characteristic equation and its three cases.

Undetermined coefficients

y″ + py′ + qy = g(x): solve the homogeneous equation, guess the form of a particular solution, and match coefficients.

Second-order initial and boundary value problems

Constant-coefficient second-order equations with initial or boundary conditions.

Spring-mass systems

Spring-mass systems: find k and m from the setup, then the motion x(t).

Variation of parameters

A particular solution by variation of parameters, when undetermined coefficients cannot work.