Vector Calculus Part I: Gradient, Divergent

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Vector Calculus Part I Gradient, Divergent Curl

• BET2533 Eng. Math. III
• R. M. Taufika R. Ismail
• FKEE, UMP

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Vector functions

• A line or curve equation in 3-D can be expressed
  as a position vector
• Position vector is often expressed in the
  parametric equation of x,y and z, that is
• if ,
  then

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Example 1

• (a) What is the line equation of a straight line
  that passes the points (1,2,2) and (2,3,-1)?
  Then sketch the line.
• (b) Sketch the graph of

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Solution

• (a) Suppose that when ,
• and when , .
  Then
• Hence,

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• (b) Parametric equation of the curve are
• Thus, we find that
• which is, the graph is the parabola
  on the plane

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Scalar and vector fields

• Imagine a cooling system of a reactor which is
  using fluid as the cooler medium

vb
va
Fluid
Tc
Td
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• At any point P, we can measure the temperature T.
  
• The temperature will depend upon whereabouts in
  the reactor we take the measurement. Of course,
  the temperature will be higher close to the
  radiator than the opening valve.
• Clearly the temperature T is a function of the
  position of the point. If we label the point by
  its Cartesian coordinates , then T
  will be a function of x, y and z, i.e.
  
• .
• This is an example of a scalar field since
  temperature is a scalar.

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• Meanwhile, at each point, the fluid will be
  moving with a certain speed in a certain
  direction
• That is, each small fluid element has a
  particular velocity and direction, depending upon
  whereabouts in the fluid it is.
• This is an example of a vector field since
  velocity is a vector. The velocity can be
  expressed as a vector function, i.e.
• where will each be scalar
  functions.

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• Physical examples of scalar fields

Electric potential around a charge
Temperature near a heated wall
(The darker region representing higher values )
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• Physical examples of vector fields

Electric field surrounding a positive and a
negative charge.
Magnetic field lines shown by iron filings
The flow field around an airplane
Hurricane
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Gradient of Scalar Fields

• The gradient of a scalar field is a vector field
  which points in the direction of the greatest
  rate of increase of the scalar field, and whose
  magnitude is the greatest rate of change.

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In the above two images, the scalar field is in
black and white, black representing higher
values, and its corresponding gradient is
represented by blue arrows.
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• Thus, the gradient of a scalar function f is
  equal to
• Even though f is a scalar, the gradient of f is a
  vector.
• The expression of grad f can be written as

where the operator is
called del or nabla.
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Example 2

• If f(x,y,z) 3x2y y2z2, find grad f and
  at the point (1,2,-1).

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Solution

• At the point (1,2,-1),

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Example 3

• Find , if
• Given

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Solution

• Since , we
  have

Integrating (1) and (2) w.r.t. x and y
respectively, we obtain
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Comparing (3) and (4), we can conclude that
and
where C is an arbitrary constant of integration
Hence,
To find constant C, use
Therefore,
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Example 4

• Find if
• and .

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Solution

• We have

Integrating (1), (2) and (3) w.r.t. x, y and z
respectively, we obtain
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Comparing (4) with (5) and (6) we get
Therefore
To find constant C, use
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Application of gradient Surface normal vector

• A normal, n to a flat surface is a vector which
  is perpendicular to that surface.
• A normal, n to a non-flat surface at a point P on
  the surface is a vector perpendicular to the
  tangent plane to that surface at P.

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• Therefore, for a non-flat surface, the normal
  vector is different, depending at the point P
  where the normal vector is located.
• Unit vector normal is defined as

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• To find the unit vector normal to the surface
  
• we follow the following
  steps
• (i) Rewrite the function as
• (ii) Find the normal vector that is
• (iii) Then, the unit normal vector is
• (iv) Hence, the unit normal vector at a point
  
• is

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Example 5

• Find the unit normal vector of the surface at
  the indicated point.
• (a) at
• (b) at

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Solution
(a) Rewrite as
Thus, we obtain
Then,
At the point (-1,3,2),
and
The unit normal vector is
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(b) Rewrite as
Thus, we obtain
Then,
At the point (1,0,-1),
and
The unit normal vector is
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Divergence of Vector Fields

• The divergence is an operator that measures the
  magnitude of a vector field's source or sink at a
  given point
• The divergence of a vector field is a scalar

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• The divergence of a vector field
• is defined as

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Curl of Vector Fields

• Curl is a vector operator that shows a vector
  field's rate of rotation, i.e. the direction of
  the axis of rotation and the magnitude of the
  rotation.

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• The curl of a vector field
• is defined as

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Example 6

• Find both div F and curl F at the point (2,0,3)
  if

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Solution
Notice that div F is a scalar!

• At the point (2,0,3),

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Notice that curl F is also a vector

• At the point (2,0,3),

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Properties of Del

• If F(x,y,z) and G(x,y,z) are differentiable
  vector functions f(x,y,z) and y(x,y,z) are
  differentiable scalar functions, then
• (i)
• (ii)
• (iii)

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• (iv)
• (v)
• (vi)
• (vii)
• (viii)

Notes In (vi), is called the Laplacian
operator