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化學數學（一） The Mathematics for Chemists (I) (Fall Term, 2004) (Fall Term, 2005) (Fall Term, 2006) Department of Chemistry National Sun Yat-sen University

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**Chapter 3 Vector Algebra and Analysis**

Definition Scalar (dot) product Vector (cross) product Scalar and vector fields Applications

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**Content covered in the textbook: Chapter 16**

Assignment: pp : 15,17,18,24,25,30,32,35,40,41, 43,47,48,55,58,60

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**Definition (naïve) B a a=AB A**

Vectors are a class of quantities that require both magnitude and direction for their specification. a Terminal point Terminal point A a=AB B Initial point Initial point Unit vector: a vector of unit length. Null vector: a vector of zero length. (its direction is meaningless.)

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**Examples of Vectors Position, velocity, angular velocity, acceleration**

Force, torque, momentum, angular momentum Electric and magnetic fields, electric and magnetic dipole moments,

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**Vector Algebra a b Equality: a b a+b Addition: a b a+b -b a-b**

Subtraction: Scalar multiplication: a 1.5a -a -0.5a

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**Example Show that the diagonals of a parallelogram bisect each other.**

We need to show that the midpoints of OD and AB coincide. O a b A B D C C’

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Classroom Exercise Show that the mean of the position vectors of the vertices of a triangle is the position vector of the centroid of the triangle. O A B C a b X B O A C a b X C’ C’’

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**Components and Decomposition**

θ a O N P x y i j a axi ayj x=(2,3),y=(4.2,-5.6)

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**Components and Decomposition (in 3D Space)**

axi azk ayj

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**Vector Algebra Restated**

Equality: Addition: Subtraction: Scalar multiplication:

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Example

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**The Center of Mass (Gravity)**

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**Dipole Moments -q r1 r Dependence of reference frame: q r2**

If the total charge Q is zero (e.g., in a molecule), then Total dipole moment:

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**Electric dipole moments**

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**Symmetry and Dipole Moment**

The total dipole moment of a tetrahedron: r4 r1=(a,a,a), r2=(a,-a,-a), r3=(-a,a,-a), r4=(-a,-a,a) r1 r2 r3

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Base Vectors Orthogonal basis: axi azk ayj Nonorthogonal basis:

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Classroom Exercise

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**Scalar Differentiation of a Vector**

B O a(t) a(t+Δt) Δa

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**Parametric Representation of a Curve**

x y z r(t) C a b t

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**Position, Velocity, Momentum, Acceleration Newton’s Second Law**

Speed: Linear momentum: Kinetic energy: Acceleration: Newton’s second law:

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Classroom Exercise Write the expression of momentum in terms of the planar polar coordinates

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**The Scalar (Dot) Product**

Proof: θ a b O B A

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Example

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θ a b O B A b b θ a θ O a O If Classroom exercise

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**Orthogonal and Coincident**

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**Cartesian Base Vectors**

Orthogonality: axi azk ayj Normalization (unit length):

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Force and Work

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**Force and Work: General Case**

B O r(t) r+Δr Δr F(r) F(r+Δr)

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**Charges in an Electric Field**

θ μ E

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**Magnetic Moment in a Magnetic Field**

θ m B

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**HOW DO YOU KNOW THEY ARE PARALLEL WITH EACH OTHER?**

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**The Vector (Cross) Product**

A new vector can be constructed from two given vectors: a b θ bsinθ Its magnitude: Its direction: v a b A B C=AxB Right-hand rule:

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**Important properties v -v Anti-commutative: b Nonassociative: a**

(Proof to be given later) Classroom exercise: If the cross product of two vectors is a zero vector, they must be parallel or antiparallel to each other.

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In Cartesian Basis axi azk ayj

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Example

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Classroom Exercise Calculate the cross product of above two vectors using

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**Application: Moment of Force (Torque)**

θ d A

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**An Electric Dipole in an Electric Field**

-q q r1 r2 O μ E T

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**A Magnetic Dipole in a Magnetic Field**

r B -qm qm r1 r2 O m B T

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Angular Velocity In a plane: v ω r O General case: θ rsinθ O r v ω

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Exercise Classroom exercise

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Exercise

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**Angular Momentum ω r p θ r O A d A special case: ω is perpendicular r:**

(moment of inertia)

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**Conservation of Angular Momentum**

If T=0, angular momentum is conserved. m B T For nuclear spins: NMR measures how fast a nuclear spin precesses.

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**Scalar and Vector Fields**

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**The Gradient of a Scalar Field**

Vector differential operator The gradient of a scalar field is a vector.

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**The Meaning of the Gradient**

f(r+dr) f(r) dr Gradient is a convenient vector expression of the derivative of multi-variable functions.

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Example: Gradient

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**Example: Gradient as Force**

Force is the negative of the gradient of the potential energy.

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**Example: Gradient as Force**

q1 q2 r q1 r E

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**The Divergence of a Vector Field**

Laplacian operator Laplace’s Equation

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**Examples: Divergence The divergence is a measure of flux density:**

the amount of ‘something’ flowing out of a unit volume per second.

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**The Curl of a Vector Field**

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Classroom Exercise

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**Physical Meaning of curl (rot)**

The curl of a velocity field is angular velocity field (x2).

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Example: Curl The curl of the velocity filed is a measure of the circulation of fluid around the point.

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**Supplementary (Not Required)**

You may win 5 points for filling in the details here!

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**Major Theorems in Integration**

Fundamental: b a L S Green: Stokes: V S Gauss:

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**Major Theorems in Integration**

b a L S V S (Stokes’s theorem) The general form: ∂M M

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Classroom Exercise? Find out the result of the following integration by using Gauss’s theorem: 1 x y z

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**The Maxwell Equations Electrostatics Electrodynamics**

Related to the light speed c In a medium:

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**Vector Spaces … Inner product space Inner (scalar) product:**

Norm (length):

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**You as a vector in a high dimensional space**

Name Gender Birth date Birth place ID Student ID Height Weight Favorite drink Favorite music …. …

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**Operations of Vectors in Fortran**

Array Loop: Dot/cross product Gradient Divergence Curl General vector spaces Subroutine

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Dot Product of Vectors Write a program to calculate the dot product of any two vectors. C Program for calculating dot product of any two vectors c program dotpro1 parameter(n1=3) real a1(n1),a2(n1) write(6,*)'Input three components of first vector:' read(5,*)(a1(i),i=1,n1) write(6,*)'Input three components of second vector:' read(5,*)(a2(i),i=1,n1) dot=0. DO 10,I = 1,n1 dot=dot+a1(i)*a2(i) CONTINUE print *,'The dot product is ', dot print *,'Next calculation (0/1)?' read(5,*)i if(i.eq.1) goto 1 stop end

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**Cross Product of Vectors**

Write a program to calculate the cross product of any two vectors. C Program for calculating cross product of any two vectors c program crossp1 parameter(n1=3) real a1(n1),a2(n1),crossp(n1) write(6,*)'Input three components of first vector:' read(5,*)(a1(i),i=1,n1) write(6,*)'Input three components of second vector:' read(5,*)(a2(i),i=1,n1) CROSSP(1)=a1(2)*a2(3)-a1(3)*a2(2) CROSSP(2)=a1(3)*a2(1)-a1(1)*a2(3) CROSSP(3)=a1(1)*a2(2)-a1(2)*a2(1) print *,'The cross product is \n', (crossp(i),i=1,3) print *,'Next calculation (0/1)?' read(5,*)i if(i.eq.1) goto 1 stop end

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**Cross Product of Vectors**

Write a program to calculate the following vector operations. C Program for calculating cross product of any three vectors program crossp2 parameter(n1=3) real a1(n1),a2(n1),a3(n1),crossp(n1) real tmp(n1) write(6,*)'Input three components of first vector:' read(5,*)(a1(i),i=1,n1) write(6,*)'Input three components of second vector:' read(5,*)(a2(i),i=1,n1) write(6,*)'Input three components of third vector:' read(5,*)(a3(i),i=1,n1) tmp1=0.0 tmp2=0.0 DO 10,I = 1,n1 tmp1=tmp1+a1(i)*a2(i) tmp2=tmp2+a1(i)*a3(i) CONTINUE do 20 i=1,n1 CROSSP(i)=tmp2*a2(i)-tmp1*a3(i) continue print *,'The cross product is \n', (crossp(i),i=1,3) print *,'Next calculation (0/1)?' read(5,*)i if(i.eq.1) goto 1 stop end

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**Gradient of a Scalar Field**

Write a program to calculate the gradient of a scalar function.

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**C Program for calculating the gradient of any scalar function**

program grdt1 parameter(n1=3) real grt(n1) write(6,*) 'please input the position (x,y,z):' read(5,*)x,y,z dx=0.001 dy=0.001 dz=0.001 x2=x+dx y2=y+dy z2=z+dz f1=x*sin(x+y)+2.0*y*z*exp(x)+3.0*z*z f2=x2*sin(x2+y)+2.0*y*z*exp(x2)+3.0*z*z fx=(f2-f1)/dx f2=x*sin(x+y2)+2.0*y2*z*exp(x)+3.0*z*z fy=(f2-f1)/dy f2=x*sin(x+y)+2.0*y*z2*exp(x)+3.0*z2*z2 fz=(f2-f1)/dz grt(1)=fx grt(2)=fy grt(3)=fz print *,'The gradient at (', x,y,z, ') is \n', (grt(i),i=1,3) print *,'Calculating the gradient of next point (0/1)?' read(5,*)i if(i.eq.1) goto 1 stop end

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**The Divergence of a Vector Field**

Write a program to calculate the divergence of a vector field.

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**C Program for calculating the divergence of any vector field**

program divg1 parameter(n1=3) real vecf(n1) write(6,*) 'please input the position (x,y,z):' read(5,*)x1,y1,z1 dx=0.0001 dy=0.0001 dz=0.0001 x2=x1+dx y2=y1+dy z2=z1+dz vx1=x1*x1 vx2=x2*x2 divgx=(vx2-vx1)/dx vy1=x1*cos(x1+x1*y1+x1*y1*z1) vy2=x1*cos(x1+x1*y2+x1*y2*z2) divgy=(vy2-vy1)/dy vz1=2.*y1+6.*z1+exp(-y1*z1) vz2=2.*y1+6.*z2+exp(-y1*z2) divgz=(vz2-vz1)/dy divg=divx+divy+divz print *,'The divergence at (', x,y,z, ') is \n', divg print *,'Calculating the gradient of next point (0/1)?' read(5,*)i if(i.eq.1) goto 1 stop end

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**The Curl of a Vector Field**

Write a program to calculate the rot of a vector field.

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**Using Subroutines (Go to Chapter 2)**

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**Cross Product of Vectors**

Write a program to calculate the following vector operations by calling subroutine for calculating cross product of two vectors.

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**C Program for calculating cross product of any two vectors**

program crossp1 parameter(n1=3) real a1(n1),a2(n1),crossp(n1) write(6,*)'Input three components of first vector:' read(5,*)(a1(i),i=1,n1) write(6,*)'Input three components of second vector:' read(5,*)(a2(i),i=1,n1) CROSSP(1)=a1(2)*a2(3)-a1(3)*a2(2) CROSSP(2)=a1(3)*a2(1)-a1(1)*a2(3) CROSSP(3)=a1(1)*a2(2)-a1(2)*a2(1) print *,'The cross product is \n', (crossp(i),i=1,3) print *,'Next calculation (0/1)?' read(5,*)i if(i.eq.1) goto 1 stop end C Subroutine for calculating cross C product of any two vectors C subroutine(a,b,c,n) real a(n),b(n),c(n) c(1)=a1(2)*a2(3)-a1(3)*a2(2) c(2)=a1(3)*a2(1)-a1(1)*a2(3) c(3)=a1(1)*a2(2)-a1(2)*a2(1) return end

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**C Program for calculating the cross**

C product of any number of vectors program crossp parameter(n=3,n1=3) real a1(n),a2(n),a3(n),TMP1(1),TMP2(N) open(20,file=‘crossp.1’,err=9999) read(20,*)(a1(i),i=1,n) read(20,*)(a2(i),i=1,n) read(20,*)(a3(i),i=1,n) close(20) call vcross(a2,a3,tmp1,n) call vcross(a1,tmp1,tmp2,n) open(30,file=‘crossp.2’,err=9999) write(30,*)(tmp2(i),i=1,n) CLOSE(30) STOP END C Subroutine for calculating cross C product of any two vectors C subroutine(a,b,c,n) real a(n),b(n),c(n) c(1)=a1(2)*a2(3)-a1(3)*a2(2) c(2)=a1(3)*a2(1)-a1(1)*a2(3) c(3)=a1(1)*a2(2)-a1(2)*a2(1) return end

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**Cross Product of Vectors**

Write a program to calculate the following vector operations by calling subroutine for calculating cross product of two vectors.

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**C Subroutine for calculating cross C product of any two vectors C **

C Program for calculating the cross C product of any number of vectors program crossp6 parameter(n=3) real a1(n),a2(n),a3(n),a4(n),a5(n),a6(n),a7(n) real TMP1(1),TMP2(N),TMP3(N),TMP4(N) real TMP5(n),TMP6(n) open(20,file=‘crossp6.1’,err=9999) read(20,*)(a1(i),i=1,n) read(20,*)(a2(i),i=1,n) read(20,*)(a3(i),i=1,n) read(20,*)(a4(i),i=1,n) read(20,*)(a5(i),i=1,n) read(20,*)(a6(i),i=1,n) read(20,*)(a7(i),i=1,n) close(20) call vcross(a6,a7,tmp1,n) call vcross(a5,a6,tmp2,n) call vcross(a3,a4,tmp3,n) call vcross(tmp1,tmp2,tmp4,n) call vcross(tmp3,tmp4,tmp5,n) call vcross(a1,tmp5,tmp6,n) open(30,file=‘crossp6.2’,err=9999) write(30,*)(tmp6(i),i=1,n) CLOSE(30) STOP END C Subroutine for calculating cross C product of any two vectors C subroutine(a,b,c,n) real a(n),b(n),c(n) c(1)=a1(2)*a2(3)-a1(3)*a2(2) c(2)=a1(3)*a2(1)-a1(1)*a2(3) c(3)=a1(1)*a2(2)-a1(2)*a2(1) return end

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General Vector Spaces Write a subroutine to calculate the inner product of two vectors in any dimension.

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**C Program for calculating dot product of any two vectors**

program dotpro1 parameter(n1=3) real a1(n1),a2(n1) write(6,*)'Input three components of first vector:' read(5,*)(a1(i),i=1,n1) write(6,*)'Input three components of second vector:' read(5,*)(a2(i),i=1,n1) dot=0. DO 10,I = 1,n1 dot=dot+a1(i)*a2(i) CONTINUE print *,'The dot product is ', dot print *,'Next calculation (0/1)?' read(5,*)i if(i.eq.1) goto 1 stop end C subroutine for calculating dot product of any two vectors subroutine vdot(a,b,n) real a(n),b(n) dot=0. DO 10,I = 1,n1 dot=dot+a(i)*b(i) CONTINUE return end

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**C program for calculating the inner product of two vectors**

program innerp1 parameter(n=10) do 10 i=1,n a1(i)=i*1.2 a2(i)=-2.*+i*i continue call vdot(a1,a2,dot,n) print *,'The inner product of two vectors is \n', dot stop end C subroutine for calculating inner product of any two vectors subroutine vdot(a,b,dot,n) real a(n),b(n) dot=0. DO 10,I = 1,n1 dot=dot+a(i)*b(i) CONTINUE return

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§1.2 Differential Calculus

§1.2 Differential Calculus

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