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Advanced Calculus TOPICS IN VECTOR CALCULUS Imam Jauhari Maknun Agustus 2017

LEARNING OUTCOME MATA KULIAH - 1  Mampu menurunkan dan menggunakan konsep dari vector calculus

dalam memecahkan masalah terapannya

TOPICS IN VECTORCALCULUS 1.

VECTOR FIELDS

2.

LINE INTEGRALS

3.

INDEPENDENCE OF PATH; CONSERVATIVE VECTOR FIELDS

4.

GREEN’S THEOREM

5.

SURFACE INTEGRALS

6.

APPLICATIONS OF SURFACE INTEGRALS; FLUX

7.

THE DIVERGENCE THEOREM

8.

STOKES’ THEOREM

VECTOR FIELDS

VECTOR FIELDS  According to Newton’s Law of Universal Gravitation, the

Earth exerts an attractive force on the mass that is directed toward the center of the Earth and has a magnitude that is inversely proportional to the square of the distance from the mass to the Earth’s center.

 This association of force vectors with points in space is called

the Earth’s gravitational field.

VECTOR FIELDS  Imagine a stream in which the water flows horizontally at

every level, and consider the layer of water at a specific depth.  At each point of the layer, the water has a certain velocity, which we can represent by a vector at that point.

 This association of velocity vectors with

points in the twodimensional layer is called the velocity field at that layer

VECTOR FIELDS  Velocity vector fields showing San Francisco Bay wind

patterns

VECTOR FIELDS

VECTOR FIELDS

VECTOR FIELDS

 In a plane xy

 In 3-space with an xyz-coordinate system

VECTOR FIELDS

Vector field on R2

Vector field on R3

GRAPHICAL REPRESENTATIONS OF VECTOR FIELDS

GRAPHICAL REPRESENTATIONS OF VECTOR FIELDS

A COMPACT NOTATION FOR VECTOR FIELDS  Sometimes it is helpful to denote the vector fields F(x, y) and

F(x, y, z) entirely in vector notation by identifying (x, y) with the radius vector r = x i + y j and (x, y, z) with the radius vector r = x i + y j + z k.  With this notation a vector field in either 2-space or 3- space

can be written as F(r).  When no confusion is likely to arise, we will sometimes omit

the r altogether and denote the vector field as F.

GRADIENT FIELDS  An important class of vector fields arises from the process of

finding gradients.  Recall that if ϕ is a function of three variables, then the gradient of ϕ is defined as :

 This formula defines a vector field in 3-space called the

gradient field of ϕ.  Similarly, the gradient of a function of two variables defines a gradient field in 2-space

GRADIENT FIELDS EXAMPLE Find the gradient field of ϕ (x, y) = x + y. Solution The gradient of ϕ is

EXERCISE 1.1 Find the gradient field of : 1. f  x, y   5 y  x3 y 2 2. f  x, y   x 2  x3 y 2  y 4 3. f  x, y    x  1 2 y  1 xy 2 4. f  x, y, z   3 z 2 2 3 5. f  x, y, z   xy  3x  z pada (2,-1,4)

6. f  x, y, z   x2 z 2 sin 4 y pada (-2,π/3,1)

CONSERVATIVE FIELDS AND POTENTIAL FUNCTIONS If F(r) is an arbitrary vector field in 2-space or 3-space, we can ask whether it is the gradient field of some function ϕ, and if so, how we can find ϕ.

CONSERVATIVE FIELDS AND POTENTIAL FUNCTIONS EXAMPLE Confirm that ϕ is a potential function for F(r)

Solution

  x, y, z   2 xi  6 yj  8 zk   F

DIVERGENCE AND CURL Divergence

DIVERGENCE AND CURL Curl (Rotational)

DIVERGENCE AND CURL EXAMPLE Find the divergence of the vector field

Solution

DIVERGENCE AND CURL EXAMPLE Find the curl of the vector field

Solution

THE ∇ OPERATOR  Gradient

 Divergence

 Curl

THE LAPLACIAN ∇2 Laplacian operator

When applied to φ(x, y, z) the Laplacian operator produces the function

Note that ∇2 ϕ can also be expressed as div (∇ ϕ).

EXERCISE 1.2 Find the divergence and the curl of the vector field

1 . F  x, y, z   xz i  yz j  xy k 2 2 2 F x , y , z  4 xy i  2 x  2 yz j  3 z  y      k 2.

3. F  x, y, z    x 2 y3  z 4  i  4 x5 y 2 z j - y 4 z 6 k

4. F  x, y, z   xe z i  4 yz 2 j + 3 y 4 e z k

LINE INTEGRALS

LINE INTEGRALS  The first goal of this section is to define what it means to

integrate a function along a curve.

LINE INTEGRALS

EVALUATING LINE INTEGRALS For a curve C in the xy-plane that is given by parametric equations x = x(t), y = y(t) (a ≤ t ≤ b)

If C is a curve in 3-space that is parametrized by x = x(t), y = y(t), z = z(t) (a ≤ t ≤ b)

EVALUATING LINE INTEGRALS EXAMPLE Evaluate the line integral from (1, 0, 0) to (−1, 0, π) along the helix C that is represented by the parametric equations x = cos t, y = sin t, z =t (0 ≤ t ≤ π)

EVALUATING LINE INTEGRALS Solution

LINE INTEGRALS WITH RESPECT TO x, y, AND z  We now describe a second type of line integral in which we

replace the “ds” in the integral by dx, dy, or dz.

LINE INTEGRALS WITH RESPECT TO x, y, AND z EXAMPLE Evaluate along the circular arc C given by x = cos t, y = sin t (0 ≤ t ≤ π/2)

LINE INTEGRALS WITH RESPECT TO x, y, AND z

EXERCISE 2.1 1. Evaluate  2 xy dx ;  2 xy dy ; C

C

 2 xy ds on the curve C defined C

by x  5 cos t ; y  5 sin t ; 0  t   4

2. Evaluate  z dx ;  z dy ;  z dz ;  z ds on the curve C defined

 by x  cos t ; y  sin t ; z  t ; 0  t  2 3. Evaluate  xy dx  x 2 dy , where C is given by y  x3 ;  1  x  2 C

C

C

C

C

4. Evaluate  y dx  x dy  z dz , C

where C is the helix x  2 cos t ; y  2 sin t ; z  t ; 0  t   4