Calculus 3 (multivariable calculus), part 1 of 2

Explore points and vectors in the plane R^2 and space R^3 using Cartesian coordinates, define position vectors, and master vector addition, scaling, and their relation to coordinates.

Learn about the four vector products—scaling, dot product, cross product, and scalar triple product—defined for R^2 and R^3, with a focus on their basics and notation.

Examine the cross product of two vectors in R^3 via a determinant-like setup, yielding a vector perpendicular to both with length equal to the parallelogram area under the right-hand rule.

Compute the scalar triple product in R^3 as (v × w) · u, a real number equal to the signed volume of the parallelepiped, and use it to test coplanarity.

Describe lines in the plane using four equivalent forms—slope-intercept, normal, intercept, and parametric—each tying a starting point, intercepts, or a directional vector.

Introduce conic sections—circle, ellipse, parabola, and hyperbola—through algebraic, geometric, distance, and analytical views, highlighting quadratic equations in two variables and their graphs.

Define conic sections by distance, showing circles, ellipses, hyperbolas, and parabolas via focal distances, directrix for parabola, and related equations and center shifts.

Present conic sections cheat sheets with circle, ellipse, and hyperbola equations, centers at x0, y0 with radius R, axes A and B, and note non-canonical forms and rotations.

Apply completing the square to determine curves. First problem yields circle centered at (1,0) with radius 1. Second yields ellipse centered at (1,0) with semi-axes 1 and 1/√2.

Identify an ellipse from a quadratic equation by completing the square to find its center at (-3, 1) and semi-axes lengths 4 and 3, yielding the canonical form.

complete the square to identify hyperbolas in problems 4 and 5. determine centers at (1,-1) and (0,1), with a and b values: problem 4, a=b=1; problem 5, a=b=1/2.

Complete the square to rewrite y = x^2 - 2x + 2 as a parabola with vertex (1,1); transform x = y^2 - 2y into a parabola with vertex (-1,1).

Explore quadric surfaces, described by second-degree equations in three variables, including ellipsoids, paraboloids, hyperboloids, cones, and cylinders, and learn to classify them via discriminants and cross-section visuals.

By completing the square, rewrite the equation as (x-3)^2+(y+2)^2-3=z, revealing an elliptic paraboloid with vertex at (3,-2,-3) and circular cross-sections.

Identify problem four's cone on the y-axis with elliptic cross-sections, and problem five's parabolic cylinder on the y-axis with cross-section x = z^2 + z.

Explore open, closed, and bounded sets in multivariable topology, learning interior, exterior, boundary, and closure with examples like ellipse domains and spheres.

Identify sets in the plane from algebraic descriptions by drawing boundary curves and using equalities and inequalities; apply these to domains of two-variable functions, double integrals, and extrema on sets.

Describe the set of points with y between -x^2+2x and 8-x^2; parabolas intersect at (4,-8), so the set is open and unbounded, not contained in a disc around the origin.

Example 3 analyzes set D in the first quadrant defined by x>0, 0<y<2, y<1/x; the set is unbounded and neither open nor closed, with boundaries x=0, y=0, y=2, y=1/x.

Example five identifies the region in the first quadrant with 1 ≤ x^2 + y^2 ≤ 4, y ≥ x, and y ≤ sqrt(3) x, which is closed and bounded.

Explore polar coordinates in the plane, using radius r and angle theta, and convert between polar and cartesian coordinates; recognize circles r = const and rays theta = const.

Convert the cylindrical equation z r = 2 - r^2 to Cartesian form using r = sqrt(x^2+y^2) and r^2 = x^2+y^2, resulting in z sqrt(x^2+y^2) = 2 - (x^2+y^2).

Explore spherical coordinates in 3d, defining points by distance from the origin and angles theta and phi, linking surfaces such as spheres and cones with r and theta/phi ranges.

Learn to convert between spherical coordinates and cartesian coordinates, deriving X, Y, Z from R, phi, theta and the inverse, including R = sqrt(X^2+Y^2+Z^2) and theta, phi relations.

Convert the Cartesian point (2, -2, 1) to spherical coordinates by computing R, theta, and phi, deriving R=3, theta=7pi/4, and phi via arccos(1/3) using the distance formula.

Convert cylindrical coordinates r=2, θ=π/6, z=-2 to cartesian and then to spherical; obtain (x,y,z) = (√3, 1, -2) and (R, φ, θ) = (2√2, 3π/4, π/6).

Explore curves from intuition to formal views in the plane and space, including graphs, conic sections, circle equations, and the helix with parametric descriptions.

Explore vector-valued functions and parameterization of curves, with X, Y, and Z components. See how different parameterizations describe the same circle, line, or ellipse, and shift center while varying speed.

Explore standard vector-valued calculus notation, including comma-separated coordinates, boldface or arrow forms, and unit axis vectors, and treat vector-valued functions as linear combinations of base vectors; avoid mixing incompatible notations.

Compute derivatives of vector-valued functions componentwise to obtain velocity and tangent vectors. Explore constant-speed circle motion, showing the velocity is perpendicular to position and derived via chain and product rules.

Compute speed as the magnitude of velocity, the derivative of position, and understand acceleration as the derivative of velocity, with circle parameterizations and chain rule implications.

Explore position, velocity, and acceleration through a concrete vector-valued example: R(t) = (t^2, sin t) for t>0, computing v=R'(t) and a=R''(t), and plotting the path as y = sin(sqrt(x)).

Sketch a parametric plane curve using a vector valued function and velocity. Determine self-intersections and turning points by analyzing x(t)=t^2-4 and y(t)=t^3-4t with t in [-2.5, 2.5].

A particle moves along a path with omega; velocity and acceleration derive, giving constant speed and a circle formed by the radius five sphere intersecting plane 3Y=4X.

Explore the extra theory of limits and continuity for vector-valued functions, and learn how component-wise convergence connects epsilon-delta definitions to two-dimensional curves.

Explore how position and velocity relate in vector-valued motion, showing velocity is perpendicular to the position vector for circular motion about the origin, and how this generalizes to sphere paths.

Compute arc length for space and plane curves by differentiating the position vector to obtain velocity and speed, then apply a change of variables to evaluate the integral.

Compute arc length for 3-d parametric curves by deriving velocity, simplifying speed with identities, and evaluating integrals from 0 to 3 and 0 to 2pi for a spiral along z-axis.