LESSON 1 Mathematics for Physics Mr. Komsilp Kotmool (Aj Tae) Department of Physics, MWIT Email : amolozo@hotmail.com Web site : www.mwit .ac.th/~tae_mwit

Warm Up the farmer needs 3 hens to produce 12 eggs in 6 days. A single chicken farmer has figured out that a hen and a half can lay an egg and a half in a day and a half. How many hens does the farmer need to produce one dozen eggs in six days? A single chicken farmer also has some cows for a total of 30 animals, and all animals in farm have 74 legs in all. How many chickens does the farmer have? the farmer needs 3 hens to produce 12 eggs in 6 days.

CONTENTS Why do we use Mathematics in Physics? What is Mathematics in this course? Functions Limits and Continuity of function Fundamental Derivative Fundamental Integration Approximation methods

Why do we use Mathematics in Physics (and other subjects)? ข้อมูลเชิงวิทยาศาสตร์ (ฟิสิกส์) Qualitative Physics – เชิงคุณภาพ เป็นรูบแบบการศึกษาที่บ่งบอกถึงความรู้สึก มีหรือไม่มี เช่น บางส่วนของวิชา Quantum Physics Quantitative Physics – เชิงปริมาณ เป็นรูปแบบการบ่งชี้เชิงข้อมูที่แม่นตรง มาก-น้อยแค่ไหน สามารถนำไปประยุกต์ใช้ในกิจกรรมของมนุษย์ได้ย่างมีประสิทธิภาพ (ส่วนใหญ่ของฟิสิกส์)

What is Mathematics in this course? REVIEW: basic problem Displacement S = ? = v x t NOT AT ALL condition: v must be constant !! But in reality, v is not constant !!! General definition Better or CALCULUS or

Functions What is FUNCTION? REVIEW : some basic equations y = Ax+B linear y = Ax2+Bx+C parabola y = sin(x) trigonometric y = circular y is the function of x for the 1-3 equation, but the 4th is not!! Rewrite: y is the function of x to be y(x) [y of x]

Functions What is FUNCTION? REVIEW again: For physical view The motion of a car is expressed with S = 5t-5t2 equation. Displacement (S) depends on time (t) We can say that S is the function of t, and can be rewritten: S(t) = 5t-5t2 Note: We usually see f(x) and g(x) in many text books.

Functions Mathematical definitions 1). If variable y depends on a variable x in such a way that each value of x determines exactly one value of y, then we say that y is a function of x. 2). A function f is a rule that associates a unique output with each input. If a input denoted by x, then the output is denoted by f(x) (read “f of x”)

Exercise Consider variable y whether it is the function or not and find the value of y at x=2 1). y – x2 = 5 2). 4y2 + 9x2 = 36 3). y + = 0 4). ycsc(x) = 1 5). y2 +2y –x =0

Limits and Continuity of function What is limits? Why do we use limits? When do we use limits? ?????

Limits and Continuity of function Newton’s Law of Universal Gravitation We can not calculate force at r → ∞. How do we solve this problem?? But we can find the close value!!!!!! at r → ∞.

Limits and Continuity of function Electric field at arbitrary from solid sphere E = ? At r = a

Limits and Continuity of function For mathematical equation f(x=0) = ? We can not find f(x) at x = 0 But we can find its close value of f(x) at x → 0 Therefore, f(x) → 2 at x → 0 “the limit of f(x) as x approaches 0 is 2”

Limits and Continuity of function How about at x → 0? -1 ; x<0 1 ; x>0 Limit from the left Limit from the right Not exist !! For discontinuity

Limits and Continuity of function THEOREMS. Let a and k be real numbers, and suppose that and 1 2 3 4 5 and

Limits and Continuity of function A function f(x) is said to be continuous at x = c provided the following condition are satisfied : 1). f(c) is defined 2). exists 3).

History of Calculus Sir Isaac Newton (1642-1727) Gottfried Wilhelm Leibniz (1646-1716)

Fundamental Derivative In reality, the world phenomena involve changing quantities: the speed of objects (rocket, vehicles, balls, etc.) the number of bacteria in a culture the shock intensity of an earthquake the voltage of an electrical signal It is very easy for the ideal situations {many exercises in your text books} For example: constant of velocity S = v x t rate of change of S is constant v = ΔS/Δt Consider: can we have this situation in the real world?

Fundamental Derivative There are many conditions in nature affecting to complicate phenomena and equations!!!

Fundamental Derivative Consider: equation of motion (free fall) S(t) = ut – ½(g)t2 Velocity does not be constant with time !!! How do we define velocity from t1 to t2 ? Average velocity ?

Fundamental Derivative Slopes and rate of change Take t2 close to t1 S(t) S(t) t t1 t2 S(t1) S(t2) Close to tangent line t1 t2 S(t1) S(t2) t vav can not be exactly represented this motion! more exactly !!! Slope at t = t1 is tangent line

Fundamental Derivative Take t2 close to t1 that t2-t1 = Δt → 0 vint → v(t) instantaneous velocity Mathematical notation (read “dee S by dee t”)

Fundamental Derivative Exercises: Find the derivative of y(x) and its value at x=2 y(x) = x y(x) = x2 y(x) = sin(x)

Fundamental Derivative FORMULARS: If f(x) and g(x) are the function of x and c is any real number 1. 2. 3. 4. 5. 6.

Fundamental Derivative What is the derivative of these complicate functions? Case 1) y(x) = sin(x2) Case 2) y(x) = xcos(x) Case 3) y(x) = tan(x) Chain rule:

Fundamental Derivative FORMULARS : If u is a function of x {u(x)}, and c is any real number Frequency use in Physics 6. 7. 1. 2. 3. 4. 5. g(x) ≠ 0

Additional applications of Derivative Maximum and Minimum problems Ex : An open box is composed of a 16-inch by 30-inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. What size should the squares to be obtained a box with the largest volume? 16 in 30 in x

Fundamental Integration What is the integration (calculus)? Why do we use the integration (calculus)? When do we use the integration (calculus)?

Fundamental Integration Recall: If we know S(t) of a object  we can find v(t) of its and say v(t) is the derivative of S(t) !! Reverse problem : If we know v(t) of a object  Can/How do we find S(t) of its ? We can say that S(t) is the antiderivative of v(t) !! What is the antiderivative?

Fundamental Integration Consider graphs of constant and linear velocity with time (v(t)-t) v(t) t v0 v S = v0t S = (1/2)(v0+v)t S(t) was represented by area under function of v(t) !!!

Fundamental Integration How about the complicate function of v(t)? v(t) t v What about S(t) of this curve? S(t) also was represented by area under of this curve! Next problem…. How do we find this area?

Fundamental Integration Method of Exhaustion The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. Archimedes used the method of exhaustion as a way to compute the area inside a circle by filling the circle with a polygon of a greater and greater number of sides. disadvantage: do not proper with asymmetric shape

Fundamental Integration v(t) t v For smoother area t* v(t*) Therefore, we get Δt Read “S(t) equal integral of v(t) dee t”

Fundamental Integration Mathematical Method Note : f(x) is the derivative of F(x), but in the other hand, F(x) is a antiderivative of f(x). For Polynomial Function F(x) = x2 f(x) = 2x Constants depend on conditions !! F(x) = x2 + 5 f(x) = 2x F(x) = x2 + 10 f(x) = 2x

Fundamental Integration Recall: Set n=n-1 Therefore, we get For the flexible form When C is the arbitrary constant.

Fundamental Integration Exercises: Find the antiderivative of the following y(x). y(x) = 3x3 + 4x2 + 5 y(x) = 0 y(x) = sin(x) y(x) = xcos(x2) y(x) = tan(x) Integration by Substitution

Fundamental Integration FORMULARS: If f(x) and g(x) are the function of x and c is any real number 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. All of these are call the indefinite integral

Fundamental Integration The definite integral Recall: We get the general solution of S(t) from the indefinite integral. Problem: How do we get displacement (S) in the interval [a,b] of time? v(t) t a b a is the lower limit b is the upper limit this is the definite integral !

Fundamental Integration Definition: (a) If a is in the domain of f(x), we define (b) If f(x) is integrable on [a,b], then we define Theorem. If f(x) is integrable on a close interval containing the three nuber a, b, and c, then

Fundamental Integration Example: Find the antiderivative of y(x) = 4x3 + 2x + 5 in the interval [2,5] of x. Sol

Fundamental Integration Exercises: Find the value of these definite integral. 1. 2. 3. 4. 5.

Additional applications of Integration In Physics Center of mass The triangle plate has mass M and constant density ρ that is shown in the figure. Find its center of mass that can find from these equations H L

Additional applications of Integration In Physics Work done by a constant force Work done by variable force ? F S

References Anton H., Bivens I., and Davis S. Calculus. 7th Ed. John Willey&Son, Inc. 2002.