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Math 2310-205

Spring 2024 Math 2310-205 Calculus III

Textbook sections covered so far

  1. Section 13.1 Vectors in the plane: notes for substitute instructor

  2. Section 13.2 Vectors in the three dimensions: notes for substitute instructor

  3. Section 13.3 Dot products: Geometric definition and algebraic definition of dot products, orthogonal projection

  4. Section 13.4 Cross products: Geometric definition and algebraic definition of cross product, Textbook Example 4, Textbook Example 3

  5. Section 12.1 Parametric equations: Textbook Example 1, MML Problem 6, Textbook Example 3(a) (MML Problem 3), Textbook Example 3(b) (MML Problems 8 and 9)

  6. Section 13.5 Lines and planes in space (Reading HW: Examples 1 or 2 lines, 5 or 6 planes, 8 or 9 parallel or orthogonal planes)
    • Textbook Example 2, MML problem 6, Textbook Example 9, MML problem 9
  7. Section 13.6 (Reading HW: Examples 1 cylinders, 2 ellipsoid, 3 elliptic paraboloid)
    • Cylinders and quadric surfaces: yellow ruled paper example y=x^2 from Textbook,
    • Textbook Example 1(a) elliptic cylinder,
    • Textbook Example 1(b) a cylinder with sin curve shape,
    • Textbook Example 2 ellipsoid,
    • Textbook Example 3 elliptic paraboloid,
    • MML Problem 9(a) and (c) figuring out an unfamiliar quadric surface using knowledge about intercepts,
    • Textbook Example 5 hyperbolic paraboloid
  8. Section 14.1 Vector-valued functions (Reading HW: Example 2 spiral, Example 3 Roller coaster or Example 4 slinky curve)
    • MML Problem 8 finding domain
    • Example 5/ MML Problem 9 Evaluating limits
    • Example 2 spiral Desmos
    • Example 4 slinky curve Desmos
  9. Section 14.2 Calculus of vector-valued functions (Review u-substitution techniques and def of tangent lines from Calc I/II. Reading HW: Example 2 unit tangent vector, Example 3 new derivative rules, Example 5 Indefinite integrals)
    • Idea of derivative/tangent vector using the vector PQ from the beginning of the section
    • Def of derivative, tangent vector, smooth, unit tangent vector
    • Example 2(a) and (b)
    • Def of antiderivative, indefinite integral, Example 6/ MML 11
    • Def of definite integral, review of integration by parts, MML 13
    • Example 3(a) using chain rule
  10. Section 14.4 Length of curves L and s(t) (Reading HW: Def of arc length for vector function, Example 2, Example 3)
    • Definition of arc length of a curve F as a definite integral of a real-valued function (i.e. a scalar function). This L is a number. Example 1 elliptical orbit
    • Definition of arc-length parametrization, the most natural parametrization: “A curve uses arc length as a parameter” if the magnitude of its derivative is 1 always.
    • Definition of the arc length function s(t). This s(t) is is a real-valued function (i.e. scalar function).
    • Example 3 (MML 6-8) finding the arc length function of a helix and using it to find a description of the curve that uses arc length as a parameter.
  11. Section 14.5 Curvature (kappa) and normal vectors (N)
    • Recall definition of unit tangent vector T from Sec 14.2
    • Idea of curvature: a nonnegative number which tells you the rate of change of direction. It does not depend on the speed of the object.
    • Definition of curvature kappa(s). This kappa(s) is always nonnegative.
    • Example 1, MML 8, Example 3
    • Idea of principal unit normal vector:
    • Definition of principal unit normal vector N. This N is a unit vector which is perpendicular to T and is pointing “inside” of the curve in the direction the curve is turning.
    • Example 4, Example 5
  12. Section 15.1 Graphs and level curves
    • Def of domain and image/range for functions of two (or more) variables
    • Example 1 domain
    • Example 2 (a), (b), (c) Graphs
    • Def of a contour curve
    • Def of a level curve
    • Example 3 (a), (b) level curves
    • Example 4 level curves
  13. Section 15.2 Limits and continuity
    • Def of limit, Theorem of limits for constant and linear functions x and y
    • Limit laws, Example 1
    • Def to memorize: interior point, boundary point, open, closed
    • Limits at boundary points: Recall from Calc 1 that limit may exist even though the function is not defined at that point. This is an example of a boundary point of the domain.
    • Example on page 933: Limit of f(x,y) as (x,y) approaches a boundary point.
    • Def of continuity, Theorem about composition of continuous functions
    • Taking limit inside a function, MML 3
  14. Section 15.3 Partial derivatives
    • Partial derivative of f(x,y) with respect to x at point (a,b) is the slope of the tangent line of the curve f(x,b). Partial derivative of f(x,y) with respect to y at point (a,b) is the slope of the tangent line of the curve f(a,y). To compute a partial derivative, treat the other variable as a constant.
    • Example 2 and 3 (partial derivatives)
    • Notation for second-order partial derivatives fxx, fyy, fxy, fyx
    • Example 4 (second-order partial derivatives)
    • Theorem: If fxy and fyx are continuous on an open set D in R^2, then fxy=fyx at all points of D
    • Example 5 (partial derivatives of a function of three variables)
    • Example 6 (Ideal Gas Law, similar to MML 10)
  15. Section 15.4 The chain rule (Reading HW: Look at the first two figures, the “tree diagrams”, the theorem on the first page, and Example 1 for chain rule with only one independent variable)
    • Chain Rule Theorem (one independent variable)
    • Example 1, geometric meaning of dz/dt
    • Chain Rule Theorem (two independent variables)
    • Example 2
    • Example 3
    • Example 4
    • Example 5 (implicit differentiation)
  16. Section 15.5 Directional derivatives and the gradient (Reading HW: Examples 1,2,3)
    • Theorem/definition of the directional derivative of f at (a,b) in the in the direction of a unit vector u. If u is i or j, then the directional derivative is a partial derivative with respect to x or y.
    • Example 1, geometric meaning of directional derivative
    • Definition of the gradient of f, as a vector-valued function
    • Example 2 (skipped, but this is a good example)
    • Example 3 (a) and (c)
    • Theorem of when rate of change is max, min, and 0:
      1. At (a,b), the direction of steepest ascent is the direction of the gradient at (a,b). The rate of change is the length of the gradient.
      2. At (a,b), the direction of steepest descent is the opposite direction of the gradient at (a,b). The rate of change is the negative of the length of the gradient.
      3. At (a,b), two directions of no change are perpendicular to the gradient at (a,b). To find vectors corresponding to these directions, use dot product.
    • Example 4
  17. Section 15.6 Tangent planes and linear approximation (Reading HW: Examples 1, 2)
    • Explicit form z=f(x,y) vs implicit form F(x,y,z)=0.
    • Definition 1: The tangent plane of the surface (in implicit form F(x,y,z)=0) at a point (a, b, c) is the plane which contains the point (a,b,c) and which has normal vector nabla F(a,b,c)
    • Example 1
    • Definition 2: The tangent plane of the surface (in explicit form z=f(x,y)) at a point (a, b, f(a,b)) is the plane which contains the point (a, b, f(a,b)) and which has normal vector <fx(a,b), fy(a,b), -1>
    • Definition: The linear approximation L(x,y) of f at (a,b) is obtained by solving for z in the Definition 2 of the tangent plane above.
    • Example 3
  18. Section 15.7 Maximum/minimum problems (Reading HW: Examples 1, 2 on critical points)
    • Definition of local extreme values, local maximum and local minimum values of f
    • Theorem: If f has a local maximum or minimum value at (a,b) and the partial derivatives exist at (a,b), then fx(a,b)=fy(a,b)=0.
    • Definition: (a,b) is a critical point if either fx(a,b)=fy(a,b)=0 OR one of the partial derivatives fx or fy does not exist at (a,b)
    • Example 1 Find critical points
    • Definition of when f has a saddle point
    • Definition of Hessian matrix and the discriminant D of f
    • Theorem (Second Derivative Test): (1) when f has a local maximum, (2) local minimum, (3) saddle point, (4) test is inconclusive
    • Definition of absolute maximum and minimum values
    • Example 5 (application, finding dimensions of a box with largest volume)
  19. Section 16.1 Double integrals over rectangular regions (Reading HW: Examples 1, 2)
    • Double integral over a rectangular region R=[a,b]x[c,d] in the xy-plane
    • Computation using iterated integral
    • Fubini theorem: order of integration can be swapped
    • Examples 1 and 2
    • Average value of f(x,y) over a region
    • MML 8
  20. Section 16.2 Double integrals over general regions (Reading HW: Examples 1, 2)
    • Double integral over a general (non-rectangular) region R in the xy-plane