# Math 2310-205

Spring 2024 Math 2310-205 Calculus III

### Textbook sections covered so far

Section 13.1 Vectors in the plane: notes for substitute instructor

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

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

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

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)

- 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

- 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

- Section 14.1 Vector-valued functions (Reading HW: Example 2 spiral, Example 3 Roller coaster or Example 4 slinky curve)
- 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

- 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.

- 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

- 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

- 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

- 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)

- 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)

- 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:
- 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.
- 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.
- 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

- 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

- 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)

- 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

- 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