Engineering Mathematics 2

Engineering Mathematics 2

Lecture 1 Yong Sung Park

Instructor

• Yong Sung Park, PhD

Associate Professor in Coastal Engineering

Dept of Civil and Environmental Engineering, SNU

• Email: dryspark@snu.ac.kr

• Office: 35-313

• Office hour: by appointment

Objectives

• Learn to use vector calculus, complex analysis, Fourier analysis and partial differential equations.

• Understand how and why those methodologies are applied in engineering subjects.

Textbook and references

• Main textbook:

• Kreyszig, E. (2011). Advanced Engineering Mathematics, 10th Ed. John Wiley & Sons (ISBN: 978-0-470-64613-7)

• References

• Hildebrand, F. B. (1976). Advanced Calculus for Applications, 2nd Ed. Prentice-Hall (ISBN: 0-13-011189-9)

• Dettman, J. W. (1984). Applied Complex Variables. Dover (ISBN: 0-486-64670-X)

• Lighthill, M. J. (1962). Introduction to Fourier Analysis and Generalised Functions.

Cambridge University (ISBN: ?)

• Carslaw, H. S. & Jaeger, J. C. (1946). Conduction of Heat in Solids. 2nd Ed. Oxford University (ISBN: 978-0-19-853368-9)

• Mei, C. C. (1989). The Applied Dynamics of Ocean Surface Waves. World Scientific (ISBN: 9971-50-773-0, 9971-50-789-7)

Lecture plan

Week No M Textbook sections

W Textbook sections

Week No M Textbook sections

W Textbook sections

1 2 Sep 9.1 – 9.4 9 26 Oct 16.3 – 16.4 28 Oct Q & A

2 7 Sep 9.5 – 9.6 9 Sep 9.7 – 10.2 10 2 Nov Exam 2

(Ch. 13 – 16)

4 Nov 11.1 – 11.3 3 14 Sep 10.3 – 10.6 16 Sep 10.7 – 10.9 11 9 Nov 11.4 – 11.6 11 Nov 11.7 – 11.8 4 21 Sep Q & A 23 Sep Exam 1

(Ch. 9 – 10)

12 16 Nov 11.9 18 Nov 12.1 – 12.4 5 28 Sep 13.1 – 13.4 30 Sep 13 23 Nov 12.5 – 12.6 25 Nov 12.7

6 5 Oct 13.5 – 13.7 7 Oct 14.1 – 14.2 14 30 Nov 12.8 – 12.9 2 Dec 12.10 7 12 Oct 14.3 – 14.4 14 Oct 15.1 – 15.2 15 7 Dec Q & A 9 Dec Exam 3

(Ch. 11 – 12) 8 19 Oct 15.3 – 15.4 21 Oct 16.1 – 16.2

Evaluation (absolute grading)

• Attendance: 10%

• Exam 1 (23 Sep): 30%

• Exam 2 (2 Nov): 30%

• Exam 3 (9 Dec): 30%

A motivating example: tsunamis in the near-shore

• multi-phase: water, sediment, trees, cars, …

• multi-scale: < 1cm ̴ 100 km

• Combination of lab experiment + analytical/numerical simulation + field data is needed.

Picture: arrival of the 2011 East Japan Tsunami

~ 100 km

Picture from Chan I.-C. & Liu, P. L.-F. (2012). JGR 117, C08006.

< 10 km

Tsunamis are extremely long

No existing wave facilities can generate such a long wave

University of Plymouth

Oregon State University flap type

piston type

A simple idea: move the whole bottom

Development of innovative 2D & 3D Wave Generators that can

generate extremely long waves.

Formulation (vector calculus and PDE)

• The veolcity potential satisfies

• with the boundary conditions

Solution by Fourier and Laplace transforms

Free-surface elevation by inverse transforms

(complex analysis)

9.1 Vectors in 2-space and 3-space

• Scalar: determined by magnitude, e.g. time, temperature, length, distance, speed, density, energy, voltage

• Vector: magnitude and direction, e.g. displacement, velocity, force

P: initial point

Q: terminal point

| Ԧ𝑎|

• unit vector: a vector of length 1

• Equality: Two vectors Ԧ𝑎 and 𝑏 are said equal, if they have the same length and the same direction.

• Translation does not change a vector.

• In a Cartesian coordinate system, a vector Ԧ𝑎 with 𝑃 𝑥₁, 𝑦₁, 𝑧₁ and 𝑄 𝑥₂, 𝑦₂, 𝑧₂ is expressed as

Ԧ

𝑎 = 𝑎₁, 𝑎₂, 𝑎₃ = 𝑥₂ − 𝑥₁, 𝑦₂ − 𝑦₁, 𝑧₂ − 𝑧₁ ,

where 𝑎_𝑖 𝑖 = 1,2,3 are called the components of the vector Ԧ𝑎, with respect to the coordinate system. Then

Ԧ

𝑎 = 𝑎₁² + 𝑎₂² + 𝑎₃².

x

y z

• Q2, Problem Set 9.1

Find the components of a vector with the initial point P (1, 1, 1) and the terminal point Q (2, 2, 0). Also find the length of the vector as well as the components of the unit vector with the same direction.

• Position vector Ԧ𝑟 of a point A (x, y, z) is the vector with A the terminal point and the origin the initial point

Ԧ𝑟 = 𝑥, 𝑦, 𝑧 .

• Vectors can be added: Ԧ𝑎 + 𝑏 = 𝑎₁ + 𝑏₁, 𝑎₂ + 𝑏₂, 𝑎₃ + 𝑏₃ .

Ԧ 𝑎

𝑏

Ԧ

𝑎 + 𝑏 𝑎Ԧ

𝑏 Ԧ 𝑎 + 𝑏

• Ԧ𝑎 + 𝑏 = 𝑏 + Ԧ𝑎 (commutative)

• Ԧ𝑎 + 𝑏 + Ԧ𝑐 = Ԧ𝑎 + 𝑏 + Ԧ𝑐 (associative)

• Ԧ𝑎 + 0 = 0 + Ԧ𝑎 = Ԧ𝑎

• Ԧ𝑎 + − Ԧ𝑎 = 0

• Scalar multiplication 𝑐 Ԧ𝑎 = 𝑐 𝑎₁, 𝑎₂, 𝑎₃ = 𝑐𝑎₁, 𝑐𝑎₂, 𝑐𝑎₃

• 𝑐 Ԧ𝑎 + 𝑏 = 𝑐 Ԧ𝑎 + 𝑐𝑏

• 𝑐 + 𝑘 Ԧ𝑎 = 𝑐 Ԧ𝑎 + 𝑘 Ԧ𝑎

• 𝑐 𝑘 Ԧ𝑎 = 𝑐𝑘 Ԧ𝑎

• 1 Ԧ𝑎= Ԧ𝑎

• 0 Ԧ𝑎 = 0

• −1 Ԧ𝑎 = − Ԧ𝑎

• ijk notation

with Ԧ𝑖 = 1,0,0 , Ԧ𝑗 = 0,1,0 , 𝑘 = 0,0,1 Ԧ

𝑎 = 𝑎₁, 𝑎₂, 𝑎₃ = 𝑎₁Ԧ𝑖 + 𝑎₂Ԧ𝑗 + 𝑎₃𝑘

9.2 Inner Product (Dot Product)

• Inner product or dot product of vectors Ԧ

𝑎 ∙ 𝑏 = Ԧ𝑎 𝑏 cos 𝛾 0 ≤ 𝛾 ≤ 𝜋

= 𝑎₁𝑏₁ + 𝑎₂𝑏₂+ 𝑎₃𝑏₃

• Non-zero vectors Ԧ𝑎 and 𝑏 are said to orthogonal to each other if Ԧ𝑎 ∙ 𝑏 = 0

Ԧ 𝑎

𝑏 𝛾

• Example 1

Find inner product of Ԧ𝑎 = [1,2,0] and 𝑏 = [3, −2,1].

Also find the length of each vector and the angle between them.

• Linear: 𝑝 Ԧ𝑎 + 𝑞𝑏 ∙ Ԧ𝑐 = 𝑝 Ԧ𝑎 ∙ Ԧ𝑐 + 𝑞𝑏 ∙ Ԧ𝑐

• Commutative: 𝑎 ∙ 𝑏 = 𝑏 ∙ ԦԦ 𝑎

• Positive definite: 𝑎 ∙ ԦԦ 𝑎 ≥ 0 and Ԧ𝑎 ∙ Ԧ𝑎 = 0 iff Ԧ𝑎 = 0

• distributive: by setting 𝑝 = 𝑞 = 1 above

• Cauchy-Schwarz inequality:

Ԧ

𝑎 ∙ 𝑏 ≤ Ԧ𝑎 𝑏

• Triangle inequality:

Ԧ

𝑎 + 𝑏 ≤ Ԧ𝑎 + 𝑏

• Parallelogram equality (show it) Ԧ

𝑎 + 𝑏 ² + Ԧ𝑎 − 𝑏 ² = 2 𝑎Ԧ ² + 𝑏 ²

• Applications of the inner product

- work 𝑊 done by force Ԧ𝑝 in the displacement of Ԧ𝑑 is 𝑊 = Ԧ𝑝 ∙ Ԧ𝑑

- component or projection of a vector Ԧ𝑎 in the direction of a vector 𝑏 𝑝 = Ԧ𝑎 ∙ ^𝑏

𝑏 = Ԧ𝑎 cos 𝛾

Ԧ 𝑎

𝛾 𝑏 Ԧ 𝑎

Ԧ

𝑎 cos 𝛾

• Example 5

Find a straight line L₁ which goes through (x, y) = (1, 3) and perpendicular to L₂: x – 2y + 2 = 0.

9.3 Vector Product (Cross Product)

• Ԧ𝑣 = Ԧ𝑎 × 𝑏, where Ԧ𝑣 = Ԧ𝑎 𝑏 sin 𝛾

• In components, Ԧ𝑣 = Ԧ𝑖 Ԧ𝑗 𝑘 𝑎₁ 𝑎₂ 𝑎₃ 𝑏₁ 𝑏₂ 𝑏₃

= Ԧ𝑖 𝑎₂𝑏₃ − 𝑎₃𝑏₂ + Ԧ𝑗 𝑎₃𝑏₁ − 𝑎₁𝑏₃ + 𝑘 𝑎₁𝑏₂ − 𝑎₂𝑏₁

y

Ԧ 𝑎

𝑏

• Q11, Problem Set 9.3

For Ԧ𝑎 = 1, −2,0 , 𝑏 = −2,3,0 , calculate Ԧ𝑎 × 𝑏 and 𝑏 × Ԧ𝑎.

• Ԧ𝑖 × Ԧ𝑗 = 𝑘, Ԧ𝑗 × 𝑘 = Ԧ𝑖, 𝑘 × Ԧ𝑖 = Ԧ𝑗

• Ԧ𝑗 × Ԧ𝑖 = −𝑘, 𝑘 × Ԧ𝑗 = −Ԧ𝑖, Ԧ𝑖 × 𝑘 = −Ԧ𝑗

• For a scalar 𝑙, 𝑙 Ԧ𝑎 × 𝑏 = 𝑙 Ԧ𝑎 × 𝑏 = Ԧ𝑎 × 𝑙𝑏

• Distributive for vector addition, Ԧ𝑎 + 𝑏 × Ԧ𝑐 = Ԧ𝑎 × Ԧ𝑐 + 𝑏 × Ԧ𝑐

• Not commutative, 𝑏 × Ԧ𝑎 = − Ԧ𝑎 × 𝑏 (anticommutative)

• Not associative, 𝑎 × 𝑏 × ԦԦ 𝑐 ≠ Ԧ𝑎 × 𝑏 × Ԧ𝑐

• Application of vector product

- Moment vector (or vector moment)

𝑚 = Ԧ𝑟 × Ԧ𝑝 in the direction of axis of rotation - Velocity of a rotating body

Ԧ

𝑣 = 𝜔 × Ԧ𝑟

r p

d 𝛾

𝜔

Ԧ𝑟 Ԧ 𝑣

• Scalar triple product Ԧ

𝑎 𝑏 Ԧ𝑐 = Ԧ𝑎 ∙ 𝑏 × Ԧ𝑐 =

𝑎₁ 𝑎₂ 𝑎₃ 𝑏₁ 𝑏₂ 𝑏₃ 𝑐₁ 𝑐₂ 𝑐₃

= Ԧ𝑎 × 𝑏 ∙ Ԧ𝑐

9.4 Vector and Scalar functions and fields

• For P, a point in a domain of definition, e.g. 3-space, surface, curve, vector function Ԧ𝑣 = Ԧ𝑣 𝑃 = 𝑣₁ 𝑃 , 𝑣₂ 𝑃 , 𝑣₃ 𝑃

• A vector function defines a vector field in the domain of definition, e.g. velocity field.

• A scalar function f = f(P) defines a scalar field in the domain, e.g. temperature field, pressure field.

• Components in a given coordinate system depend on the coordinate system.

• However vector function and scalar function and their fields do NOT depend on a coordinate system.

• Limit vector of a sequence of vectors Ԧ

𝑎 = lim

𝑛→∞ 𝑎Ԧ _𝑛

• A limit of a vector function of a real variable Ԧ𝑙 = lim

𝑡→𝑡₀ 𝑣 𝑡Ԧ

• A vector function Ԧ𝑣(𝑡) is said continuous at 𝑡 = 𝑡₀, if it is defined in the neighborhood of 𝑡₀, and lim

𝑡→𝑡₀ 𝑣 𝑡 = ԦԦ 𝑣(𝑡₀)

• A vector function Ԧ𝑣(𝑡) is said differentiable if the limit Ԧ

𝑣^′ 𝑡 = lim

∆𝑡→0

𝑣 𝑡+∆𝑡 −𝑣(𝑡)

∆𝑡 exists.

• Ԧ𝑣′(𝑡) is called the derivative of Ԧ𝑣(𝑡) and in a Cartesian coordinate Ԧ

𝑣^′ 𝑡 = 𝑣₁^′ 𝑡 , 𝑣₂^′ 𝑡 , 𝑣₃′(𝑡)

• Therefore the usual differentiation rules apply to the vector functions.

• Q24, Problem Set 9.4

Find the first partial derivatives Ԧ𝑣 𝑥, 𝑦 = 𝑒^𝑥 cos 𝑦, 𝑒^𝑥 sin 𝑦 .