Interactive Laboratories

Calculus III for Engineers

3D labs, unit by unit

Orbit, drag and slice. Every lab names the lesson section that explains it, one click away.

Unit 1 · Vectors & geometry

Vector bench

Vectors in 3D

Drag two vectors through space; the dot product, cross product, sum and difference respond as you move them.

Orbit viewDot & cross Vector bench

Into & out of the page

Look straight down the z-axis and see what the sign of a 2D cross product actually means.

Right-hand rule Reference

Vector formula sheet

Every readout on the vector benches, written out as the equations behind it.

Formulas 3D lab · Unit 1

Plane explorer

Set a, b, c, d in ax + by + cz = d and watch the normal, the intercepts and the point-to-plane distance.

PlanesDistance 3D lab · Unit 1

Quadric surface viewer

Slide a trace plane through each quadric and name the curve it cuts — the exam identification, done live.

TracesSurfaces 3D lab · Unit 1

Locate a point three ways

One point, three constructions: Cartesian, cylindrical and spherical, with the conversions worked below.

Coordinates

Unit 2 · Vector functions

3D lab · Unit 2

Space curve & moving frame

Ride a curve with T, N and B attached, and watch the osculating circle track the curvature.

TNB frameCurvature

Unit 3 · Partial derivatives

3D lab · Unit 3

Surface, contours & gradient

The same function as a surface and a contour map; the gradient stays perpendicular to the contours and points uphill.

GradientDirectional derivative 3D lab · Unit 3

Critical points & the second-derivative test

Tilt the tangent plane until it goes flat, then read the Hessian to tell a bowl from a saddle.

HessianExtrema

Unit 4 · Multiple integrals

3D lab · Unit 4

Riemann boxes under a surface

Slide n up and watch the staircase of boxes melt into the surface, with the sum tracked against the exact value.

Double integrals 3D lab · Unit 4

Region explorer: strips & limits

Sweep a strip across a region and read the limits off its ends — then switch the order and see a new description.

Order of integration 3D lab · Unit 4

The volume element, edge by edge

A wedge of space with its three edges labelled, showing where r and ρ² sin φ come from.

CylindricalSpherical

Unit 5 · Vector fields

3D lab · Unit 5

Work along a path

Pick a field and a path, drag the endpoints, and see ∫_C F·dr computed as you move.

Line integralsConservative fields 3D lab · Unit 5

Divergence & curl probe

Drag a probe through the field; a shrinking circle measures flux and circulation, and a paddle wheel spins with the curl.

DivergenceCurl

Unit 6 · Surface integrals

3D lab · Unit 6

Flux through a surface

Normals coloured by whether the field leaves or enters, with the flux computed numerically.

FluxDivergence theorem 3D lab · Unit 6

Stokes checker

One boundary circle, four different caps — and both sides of Stokes’ theorem agreeing every time.

Stokes’ theorem

All labs

Unit 1 · Vectors & geometry · vector bench

Two vectors in space

A and B start at the origin. Drag the view to orbit it, drag an arrowhead to move that vector in the plane of the screen, or use the sliders for exact components. The graph on the right shows how the two products would change if only the angle between them changed.

A B A + B A − B A × B projection of A on B

Drag empty space to orbit. Drag an arrowhead to move that vector. ⊙ means a vector pointing straight at you, ⊗ straight away.

A·B = |A||B| cos θ and |A×B| = |A||B| sin θ against θ. The solid dot is where you are; the hollow one is the other angle with the same sine.

Show

Vector A

A_x

A_y

A_z

Vector B

B_x

B_y

B_z

Presets

Motion

Speed 40 °/s

Zoom 1.00×

|A|—

|B|—

θ between—

θ in radians—

A · B—

cos θ—

θ from arccos—

A along B—

A × B—

|A × B|—

sin θ—

θ from arcsin—

A + B—

|A + B|—

A − B—

|A − B|—

Why arccos is safe and arcsin is not

Between two vectors θ runs from 0° to 180°. Cosine is different for every angle in that range, so A·B pins θ down uniquely. Sine is not: sin 60° = sin 120°, so |A×B| alone can only tell you “60° or 120°”. Look at the hollow dot on the graph — it never goes away. The sign of the dot product breaks the tie: positive means the acute answer, negative means the obtuse one.

Things worth trying

Pick Perpendicular. A·B reads exactly 0, the projection collapses to a point, and |A×B| is as big as it can be for those lengths — |A||B|.
Pick Parallel, then Antiparallel. The cross product vanishes both times; only the sign of the dot product tells them apart.
Choose Sweep θ and press Play. The cross product stays on one axis the whole time, growing and shrinking, and flips to the opposite side the instant B passes through A’s line.
Choose Spin B about A. The angle and the dot product freeze, the projection freezes, and A×B sweeps out a circle — everything about the cross product is direction.
Switch to the Top-down view with the Flat preset. A×B turns into a ⊙ symbol: it points straight at you, out of the page.
Turn on A − B and drag things around. The difference always runs from the tip of B to the tip of A — that is the arrow you would add to B to reach A.

Unit 1 · Vectors & geometry · vector bench

Into and out of the page

The same A and B, now seen from directly above the xy-plane — this is the “page” in every textbook diagram, with z pointing at your face. Only the x and y components matter here, so the cross product is purely along z: A_xB_y − A_yB_x. Positive means out of the page toward you (⊙); negative means into the page (⊗). Flip the viewpoint and both the symbol and the rotation sense reverse — they are the same physical fact seen from the other side.

Viewed from

Drag the arrowheads. Only x and y change here — z is left alone.

(A × B)_z—

Rotation A → B—

A × B points—

θ in the plane—

Right-hand rule

Point the fingers of your right hand along A, curl them toward B through the smaller angle, and your thumb gives A×B. Counterclockwise curl (seen from above) → thumb up → +z → out of the page. Clockwise → thumb down → −z → into the page. Swap the order and B×A is the same length pointing the other way.

Things worth trying

Drag B across the line of A. The instant it crosses, the arc flips direction and the symbol flips from ⊙ to ⊗ — the cross product has no “small” way to change sign.
Switch to Below. The arc that read counterclockwise now reads clockwise, and the symbol flips — but (A×B)_z in the readout does not change. Into/out of the page is about where you are standing; the vector never moved.
Make the two vectors collinear. The arc disappears and (A×B)_z reads 0 — no rotation means no cross product.

Unit 1 · Vectors & geometry · reference

Formula sheet

Everything the readouts above are computing, in one place.

\[ \vec A + \vec B = (A_x + B_x,\; A_y + B_y,\; A_z + B_z) \qquad \vec A - \vec B = (A_x - B_x,\; A_y - B_y,\; A_z - B_z) \] Componentwise. Geometrically, put the tail of \(\vec B\) on the tip of \(\vec A\) for the sum; \(\vec A - \vec B\) runs from the tip of \(\vec B\) to the tip of \(\vec A\).

\[ \vec A \cdot \vec B = A_xB_x + A_yB_y + A_zB_z = |\vec A|\,|\vec B|\cos\theta \] \[ \theta = \arccos\frac{\vec A\cdot\vec B}{|\vec A|\,|\vec B|} \qquad \text{scalar projection of } \vec A \text{ on } \vec B = \frac{\vec A\cdot\vec B}{|\vec B|} \] A scalar. Positive when the angle is acute, zero when perpendicular, negative when obtuse. Because \(\cos\) is one-to-one on \([0^\circ, 180^\circ]\), arccos returns the angle without ambiguity.

\[ \vec A \times \vec B = \begin{vmatrix} \hat\imath & \hat\jmath & \hat k \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix} = (A_yB_z - A_zB_y)\,\hat\imath + (A_zB_x - A_xB_z)\,\hat\jmath + (A_xB_y - A_yB_x)\,\hat k \] \[ |\vec A \times \vec B| = |\vec A|\,|\vec B|\sin\theta \qquad \theta = \arcsin\frac{|\vec A\times\vec B|}{|\vec A|\,|\vec B|}\;\;\text{(returns only } 0^\circ\text{–}90^\circ) \] A vector, perpendicular to both inputs, direction from the right-hand rule, and anticommutative: \(\vec B\times\vec A = -\,\vec A\times\vec B\). Its magnitude is the area of the parallelogram the two vectors span. Since \(\sin\theta = \sin(180^\circ-\theta)\), the arcsin route needs the sign of the dot product to decide between the two.

\[ \text{For } \vec A, \vec B \text{ in the } xy\text{-plane:}\qquad \vec A\times\vec B = (A_xB_y - A_yB_x)\,\hat k \] Positive \(\Rightarrow\) \(+z\), out of the page, \(\vec A\to\vec B\) counterclockwise (viewed from \(+z\)). Negative \(\Rightarrow\) \(-z\), into the page, clockwise.

Unit 1 · Vectors & geometry

Plane explorer

The sliders set \(a, b, c, d\) in \(ax + by + cz = d\). The orange arrow is the normal \(\mathbf{n} = \langle a,b,c\rangle\), the dots on the axes are the intercepts, and the dashed segment drops from the free point \(Q\) perpendicularly onto the plane — its length is the distance formula. Drag the view to orbit. Try making \(c = 0\) and watch the plane stand up vertically, parallel to the \(z\)-axis.

Presets

Drag to orbit. Orange arrow: normal \(\mathbf{n}\). Blue dot: \(Q\). Dashed: perpendicular from \(Q\) to the plane.

View

Plane \(ax + by + cz = d\)

a1.0

b1.0

c1.0

d2.0

Free point \(Q\)

Q_x2.0

Q_y2.0

Q_z2.0

Equation—

|n|—

Distance from origin—

Distance from Q—

Intercepts x, y, z—

Distance from Q, live

Theory for this lab: Unit 1 lesson · Planes →

Unit 1 · Vectors & geometry

Quadric surface viewer

Pick a surface, adjust \(a, b, c\), then slide the trace plane through it. The highlighted curve is the trace in that plane, and the readout names it — exactly the identification you would do by hand on an exam. Notice how the hyperboloids are the cone with the tip pulled apart (one sheet) or pinched open (two sheets).

Drag to orbit. The translucent sheet is the trace plane; the orange curve is the trace.

Trace plane

k0.50

Shape parameters

a1.00

b1.00

c1.00

Show

Surface—

Trace—

Equation with your a, b, c

Theory for this lab: Unit 1 lesson · Quadric surfaces →

Unit 1 · Vectors & geometry

Locate a point three ways

One point, three descriptions. Choose which system you want to drive with the chips, move the sliders, and watch the same point being located by a different construction: three perpendicular steps (Cartesian), a radius and a turn in the floor plus a rise (cylindrical), or a distance and two angles (spherical). The conversions are worked live below the readouts with your numbers in them.

Drive with

Drag to orbit. Dashed lines are the Cartesian steps; the foot on the floor is the point's shadow.

x1.50

y2.00

z1.50

Show

View

Cartesian (x, y, z)—

Cylindrical (r, θ, z)—

Spherical (ρ, θ, φ)—

|OP|—

Conversions with your numbers

Theory for this lab: Unit 1 lesson · Cylindrical and spherical coordinates →

Unit 2 · Vector functions

Space curve & moving frame

A point rides along the curve; the three arrows are \(\mathbf{T}\), \(\mathbf{N}\) and \(\mathbf{B}\) at that point. The dashed circle is the osculating circle — watch it shrink where the curve bends hard and blow up where the curve is nearly straight. The graph on the right plots \(\kappa\) and speed against \(t\). Compare the helix (both constant) with the twisted cubic (neither constant) — then press play.

T N B velocity acceleration

Drag to orbit. The solid part of the curve has been traversed; the point moves in the direction of increasing \(t\).

κ(t) and speed |r′(t)| along the curve. The vertical line is where you are.

Parameter t0.00

Show

Speed1.0×

r(t)—

Speed |r′|—

Curvature κ—

Radius 1/κ—

a_T (tangential)—

a_N = κv²—

Curvature, live

Theory for this lab: Unit 2 lesson · Curvature and the TNB frame →

Unit 3 · Partial derivatives

Surface, contour map & gradient

The same function drawn twice: as a surface on the left and as a contour map on the right. Drag the point on the map (or use the sliders). The orange arrow is \(\nabla f\) at that point — watch it stay perpendicular to the contour and point uphill. The blue arrow is a direction \(\mathbf{u}\) you control with the angle slider; the tilted blue line on the surface has slope \(D_{\mathbf{u}}f\). Rotate \(\mathbf{u}\) to line up with the gradient and the slope hits its maximum, \(|\nabla f|\); rotate it along the contour and the slope goes to zero.

Drag to orbit. The contour through \(P\) is drawn on the surface; \(\nabla f\) and \(\mathbf{u}\) are drawn on the floor beneath \(P\).

Drag the point. Orange: \(\nabla f\). Blue: \(\mathbf{u}\). Contours every fixed step of \(f\); the bold one passes through \(P\).

x₀0.60

y₀−0.40

φ of u30°

Show

Point u

f(x₀, y₀)—

f_x—

f_y—

|∇f| (max slope)—

Direction of ∇f—

D_u f—

Directional derivative, live

Theory for this lab: Unit 3 lesson · The gradient and directional derivatives →

Unit 3 · Partial derivatives

Critical points & the second-derivative test

The translucent square is the tangent plane at \(P\). Move \(P\) around and watch it tilt; at a critical point it goes flat. Then read the Hessian: on the bowl, \(D > 0\) with \(f_{xx} > 0\); on the saddle, \(D < 0\), and the flat tangent plane cuts through the surface. The cubic preset has one minimum and one saddle side by side — use Next critical point to jump between them and compare the numbers. The monkey saddle is the \(D = 0\) case the test cannot decide.

Drag to orbit. Tangent plane at \(P\) in violet; contours of \(f\) on the floor below.

x₀0.80

y₀0.60

Show

Function

f_x—

f_y—

f_xx—

f_yy—

f_xy—

D = f_xxf_yy − f_xy²—

Verdict

Theory for this lab: Unit 3 lesson · Maxima, minima and saddle points →

Unit 4 · Multiple integrals

Riemann boxes under a surface

The square \([0,2]\times[0,2]\) is cut into \(n\times n\) tiles and a box is built on each one, with height equal to \(f\) at the chosen sample point. Slide \(n\) up and watch the staircase melt into the surface; the graph on the right tracks the sum against \(n\) and the exact value. Note how much faster the midpoint rule converges than either corner rule — the corner rules are consistently biased on a monotone surface, the midpoint rule is not.

Drag to orbit. Boxes coloured by height; the faint mesh is the true surface.

Riemann sum against \(n\) for the current rule; dashed line is the exact integral. The dot is your current \(n\).

Subdivisions n4 × 4

Sample point

Show

ΔA = Δx·Δy—

Riemann sum—

Exact ∬ f dA—

Error—

The sum being computed

Theory for this lab: Unit 4 lesson · Double integrals over rectangles →

Unit 4 · Multiple integrals

Region explorer: strips, limits and order

Pick a region and a strip direction. The highlighted strip slides across the region; the labels at its ends are the inner limits, and the integral below is written out for that order with the limits filled in. Switch the direction and watch the same region get a completely different description. The last preset needs two integrals in one direction and only one in the other.

Vertical strips: \(dy\) inside, \(dx\) outside. The strip enters through the lower curve and leaves through the upper one.

Strips

Strip—

The region

Iterated integral in this order

Strip at—

Inner limits here—

Strip length—

Area ∬ 1 dA (either order)—

Theory for this lab: Unit 4 lesson · General regions and the order of integration →

Unit 4 · Multiple integrals

The volume element, edge by edge

A small wedge of space at the position you choose, sitting on its coordinate surface (a cylinder for cylindrical, a sphere for spherical). Its three edges are labelled with their lengths. Move it outward and watch the arc edges grow with \(r\) or \(\rho\); in spherical, move it toward the pole and watch the \(\theta\)-edge shrink with \(\sin\phi\). The readout compares the product of the three edges with the wedge's exact volume — they agree in the limit of small \(\Delta\), which is all a differential ever claims.

Drag to orbit. The faint surface is \(r = \) const (cylindrical) or \(\rho = \) const (spherical). Edge labels are lengths.

r1.60

θ40°

z0.40

Δ size0.45

Show

Edge 1: Δr—

Edge 2: r Δθ—

Edge 3: Δz—

Product of edges—

Exact wedge volume—

Ratio—

Volume element with your numbers

Theory for this lab: Unit 4 lesson · Triple integrals in cylindrical and spherical coordinates →

Unit 5 · Vector fields

Work along a path through a field

Choose a field and a path, then drag the endpoints. The integral \(\int_C\mathbf{F}\cdot d\mathbf{r}\) is computed numerically as you move; the graph on the right shows \(\mathbf{F}\cdot\mathbf{T}\) along the path, and the shaded area is the work. The experiment to run: pick a conservative field, keep \(A\) and \(B\) fixed, and switch between the straight segment, the arc and the detour — the work does not change. Pick the shear or rotational field and it does. Then try the closed loops.

Field

Path

Drag \(A\) or \(B\). Arrows show the field; the orange path is \(C\) with its direction marked.

\(\mathbf{F}\cdot\mathbf{T}\) against arc length \(s\). Shaded area (sign included) is \(\int_C\mathbf{F}\cdot d\mathbf{r}\).

∫_C F·dr—

Path length—

f(B) − f(A) if F = ∇f—

Field—

This field

Theory for this lab: Unit 5 lesson · Line integrals →

Unit 5 · Vector fields

Divergence & curl probe

Drag the probe anywhere in the field. The small circle around it measures two things numerically: the net outward flux through the circle (divided by its area, this approaches \(\nabla\cdot\mathbf{F}\)) and the circulation around it (divided by its area, this approaches the curl). The paddle wheel spins at the rate the curl predicts. Shrink the circle and watch the estimates converge to the analytic values. The vortex field is the one to think about: the wheel does not turn anywhere — except that a circle around the origin still has circulation \(2\pi\).

Field

Drag the probe. Green arrows: outward component of \(\mathbf{F}\) on the circle (flux). Violet: tangential component (circulation).

Radius ε0.50

x1.00

y0.60

Show

This field

Flux out ÷ area—

∇·F at the probe—

Circulation ÷ area—

(∇×F)_z at the probe—

Theory for this lab: Unit 5 lesson · Curl and divergence →

Unit 6 · Surface integrals

Flux through a surface

Pick a surface and a field. Short arrows mark the surface normals, coloured green where the field goes out through the surface (\(\mathbf{F}\cdot\mathbf{n} > 0\)) and amber where it goes in. The flux is computed numerically. For closed surfaces the readout also shows \(\iiint\nabla\cdot\mathbf{F}\,dV\) over the inside — Section 5 says they must agree. Try: rotational field, any closed surface (zero flux — the field only circulates); inverse-square, sphere then cube (both \(4\pi\)); uniform field, hemisphere then disk (equal, because the two share a boundary and the field has no divergence).

Surface

Field

Drag to orbit. Normals: green = field leaving, amber = field entering, grey = tangent.

Size R1.20

Show

This field

Flux ∬ F·dS—

Surface area—

∭ ∇·F dV inside (closed only)—

Note—

Theory for this lab: Unit 6 lesson · Orientation and flux →

Unit 6 · Surface integrals

Stokes checker: one boundary, four caps

The orange circle is \(C\), the unit circle in the \(xy\)-plane traversed counterclockwise seen from above. Each cap has that circle as its boundary and is oriented upward, as the right-hand rule requires. The lab computes both sides of Stokes' theorem numerically: the circulation of \(\mathbf{F}\) around \(C\), and the flux of \(\nabla\times\mathbf{F}\) through whichever cap you choose. Switch caps — the flux of the curl does not change, because the boundary did not.

Cap

Field

Drag to orbit. Orange: the boundary and \(\mathbf{F}\) along it. Violet: \(\nabla\times\mathbf{F}\) on the cap.

Show

Field and its curl

Both sides

∮_C F·dr—

∬_S (∇×F)·dS—

Difference—

Cap area—

Theory for this lab: Unit 6 lesson · Stokes' theorem →