# Math 251

Engineering Mathematics II
Joe Kahlig

## Lecture Notes

The class notes contain the concepts and problems to be covered during lecture. Printing and bringing a copy of the notes to class will allow you to spend less time trying to write down all of the information and more time understanding the material/problems. Additional examples may be included during the lectures to clarify/illustrate concepts.

As an alternate to taking notes on paper, consider taking notes on your laptop using OneNote, or a similar program.

Completed notes will be posted after the section has been finished. Be advised that the notes may not included everything that is said during class.
The problems will be completely worked out but some additional information (hints, warnings, analysis of the problem,..) may not we written out in the completed notes.

Exam 1 Material
Section Notes Solutions videos
12.1: Three Dimensional Coordinate Systems blank completed video
video pdf
12.2: Vectors blank completed video
video pdf
12.3: The Dot Product blank completed video
video pdf
12.4: The Cross Product blank completed part a
part b
video pdf
12.5: Equations of Lines and Planes blank completed part a
part b
part c
video pdf
blank completed video
video pdf
13.1: Vector Functions and Space Curves blank completed part a
part b
video pdf
13.2: Derivative and Integrals of Vector Functions blank completed video
video pdf
13.3: Arc Length and Curvature blank completed part a
part b
video pdf
13.4: Motion in Space: Velocity and Acceleration blank completed video
video pdf

Exam 2 Material
Section Notes Solutions videos
14.1: Functions of Several Variables blank completed part a
part b
video pdf
14.3: Partial Derivatives blank

#### The lecture notes below this yellow bar are in the process of being updated.

14.4: Tangent Planes and Differentials blank
14.5: The Chain Rule blank
14.6: Directional Derivatives and the Gradient Vector blank
14.7: Maximum and Minimum Values blank
14.8: Lagrange Multipliers blank

Exam 3 Material
Section Notes Solutions videos
10.3: Review of Polar Coordinates blank
15.1: Double Integrals over Rectangles blank
15.2: Double Integrals over General Regions
Math 152 lecture notes on Integration: see 7.1, 7.2, 7.3, 7.4
blank
15.3: Double Integrals in Polar Coordinates blank
15.4: Applications of Double Integrals blank
15.5: Surface Area blank
15.6: Triple Integrals blank
15.7: Triple Integrals in Cylindrical Coordinates blank
15.8: Triple Integrals in Spherical Coordinates blank
15.9: Change of Variables blank

Final Exam Material
Section Notes Solutions videos
16.1: Vector Fields blank
16.2: Line Integrals blank
16.3: Fundamental Theorem for Line Integrals blank
16.4: Green's Theorem blank
16.5: Curl and Divergence blank
16.6: Parametric Surfaces and Their Areas blank
16.7: Surface Integrals blank
16.8: Stokes' Theorem blank
16.9: Divergence Theorem blank