3 examples of PDE, for Laplace, Diffusion of Heat and Wave function. A brief definition of Fouriers Series. Slides created and compiled using LaTeX, beamer package.
Application of partial derivatives with two variablesSagar Patel
Application of Partial Derivatives with Two Variables
Maxima And Minima Values.
Maximum And Minimum Values.
Tangent and Normal.
Error And Approximation.
3 examples of PDE, for Laplace, Diffusion of Heat and Wave function. A brief definition of Fouriers Series. Slides created and compiled using LaTeX, beamer package.
Application of partial derivatives with two variablesSagar Patel
Application of Partial Derivatives with Two Variables
Maxima And Minima Values.
Maximum And Minimum Values.
Tangent and Normal.
Error And Approximation.
Changing variable is something we come across very often in Integration. There are many
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Changing variable is something we come across very often in Integration. There are many
reasons for changing variables but the main reason for changing variables is to convert the
integrand into something simpler and also to transform the region into another region which is
easy to work with. When we convert into a new set of variables it is not always easy to find the
limits. So, before we move into changing variables with multiple integrals we first need to see
how the region may change with a change of variables. In order to change variables in an
integration we will need the Jacobian of the transformation.
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry.
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3. INTRODUCTION
If f is a function of two variables x and y, suppose we
let only x vary while keeping y fixed, say y = b, where b
is a constant.
Then we are really considering a function of a single
variable x, namely, g(x) = f(x, b).
If g has a derivative at a, then we call it the partial
derivative of f with respect to x at (a, b) and denote it
by fx(a, b).
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5. INTRODUCTION (CONT’D)
Similarly, the partial derivative of f with respect
to y at (a, b), denoted by fy(a, b), is obtained by
holding x = a and finding the ordinary derivative
at b of the function G(y) = f(a, y):
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6. DEFINITION
If we now let the point (a, b) vary, fx and fy
become functions of two variables:
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7. NOTATIONS
There are many alternate notations for partial
derivatives:
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8. FINDING PARTIAL DERIVATIVES
The partial derivative with respect to x is just the
ordinary derivative of the function g of a single
variable that we get by keeping y fixed:
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9. EXAMPLE
If f(x, y) = x3 + x2y3 – 2y2, find fx(2, 1) and fy(2, 1).
Solution Holding y constant and differentiating
with respect to x, we get
fx(x, y) = 3x2 + 2xy3
and so
fx(2, 1) = 3 · 22 + 2 · 2 · 13 = 16
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10. SOLUTION (CONT’D)
Holding x constant and differentiating with
respect to y, we get
fx(x, y) = 3x2y2 – 4y
and so
fx(2, 1) = 3 · 22 · 12 – 4 · 1 = 8
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11. INTERPRETATIONS
To give a geometric interpretation of partial
derivatives, we recall that the equation z = f(x, y)
represents a surface S.
If f(a, b) = c, then the point P(a, b, c) lies on S. By
fixing y = b, we are restricting our attention to
the curve C1 in which the vertical plane y = b
intersects S.
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12. INTERPRETATIONS (CONT’D)
Likewise, the vertical plane x = a intersects S in
a curve C2.
Both of the curves C1 and C2 pass through the
point P.
This is illustrated on the next slide:
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14. INTERPRETATIONS (CONT’D)
Notice that…
C1 is the graph of the function g(x) = f(x, b), so the
slope of its tangent T1 at P is g′(a) = fx(a, b);
C2 is the graph of the function G(y) = f(a, y), so the
slope of its tangent T2 at P is G′(b) = fy(a, b).
Thus fx and fy are the slopes of the tangent lines
at P(a, b, c) to C1 and C2.
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15. INTERPRETATIONS (CONT’D)
Partial derivatives can also be interpreted as
rates of change.
If z = f(x, y), then…
∂z/∂x represents the rate of change of z with respect
to x when y is fixed.
Similarly, ∂z/∂y represents the rate of change of z
with respect to y when x is fixed.
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16. EXAMPLE
If f(x, y) = 4 – x2 – 2y2, find fx(1, 1) and
fy(1, 1).
Interpret these numbers as slopes.
Solution We have
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17. SOLUTION (CONT’D)
The graph of f is the paraboloid
z = 4 – x2 – 2y2 and the vertical plane y = 1
intersects it in the parabola z = 2 – x2, y = 1.
The slope of the tangent line to this parabola at
the point (1, 1, 1) is fx(1, 1) = –2.
Similarly, the plane x = 1 intersects the graph of f
in the parabola z = 3 – 2y2, x = 1.
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18. SOLUTION (CONT’D)
The slope of the tangent line to this parabola at the
point (1, 1, 1) is fy(1, 1) = –4:
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19. EXAMPLE
If
Solution Using the Chain Rule for functions of
one variable, we have
.andcalculate,
1
sin,
y
f
x
f
y
x
yxf
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20. EXAMPLE
Find ∂z/∂x and ∂z/∂y if z is defined implicitly as a
function of x and y by
x3 + y3 + z3 + 6xyz = 1
Solution To find ∂z/∂x, we differentiate implicitly with
respect to x, being careful to treat y as a constant:
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21. SOLUTION (CONT’D)
Solving this equation for ∂z/∂x, we obtain
Similarly, implicit differentiation with respect to y
gives
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22. MORE THAN TWO VARIABLES
Partial derivatives can also be defined for functions of
three or more variables, for example
If w = f(x, y, z), then fx = ∂w/∂x is the rate of change of
w with respect to x when y and z are held fixed.
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23. MORE THAN TWO VARIABLES (CONT’D)
But we can’t interpret fx geometrically because the
graph of f lies in four-dimensional space.
In general, if u = f(x1, x2 ,…, xn), then
and we also write
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24. EXAMPLE
Find fx , fy , and fz if f(x, y, z) = exy ln z.
Solution Holding y and z constant and
differentiating with respect to x, we have
fx = yexy ln z
Similarly,
fy = xexy ln z and fz = exy/z
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25. HIGHER DERIVATIVES
If f is a function of two variables, then its partial
derivatives fx and fy are also functions of two
variables, so we can consider their partial
derivatives
(fx)x , (fx)y , (fy)x , and (fy)y ,
which are called the second partial derivatives of
f.
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27. EXAMPLE
Find the second partial derivatives of
f(x, y) = x3 + x2y3 – 2y2
Solution Earlier we found that
fx(x, y) = 3x2 + 2xy3 fy(x, y) = 3x2y2 – 4y
Therefore
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28. MIXED PARTIAL DERIVATIVES
Note that fxy = fyx in the preceding example,
which is not just a coincidence.
It turns out that fxy = fyx for most functions that
one meets in practice:
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29. MIXED PARTIAL DERIVATIVES (CONT’D)
Partial derivatives of order 3 or higher can also be
defined. For instance,
and using Clairaut’s Theorem we can show that fxyy =
fyxy = fyyx if these functions are continuous.
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31. PARTIAL DIFFERENTIAL EQUATIONS
Partial derivatives occur in partial differential
equations that express certain physical laws.
For instance, the partial differential equation
is called Laplace’s equation.
02
2
2
2
y
u
x
u
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32. PARTIAL DIFFERENTIAL EQNS.
(CONT’D)
Solutions of this equation are called harmonic
functions and play a role in problems of heat
conduction, fluid flow, and electric potential.
For example, we can show that the function u(x,
y) = ex sin y is a solution of Laplace’s equation:
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34. PARTIAL DIFFERENTIAL EQNS.
(CONT’D)
The wave equation
describes the motion of a waveform, which could
be an ocean wave, sound wave, light wave, or
wave traveling along a string.
2
2
2
2
2
x
u
a
t
u
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35. PARTIAL DIFFERENTIAL EQNS.
(CONT’D)
For instance, if u(x, t) represents the
disaplacement of a violin string at time t and at a
distance x from one end of the string, then u(x, t)
satisfies the wave equation. (See the next slide.)
Here the constant a depends on the density of the
string and on the tension in the string.
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37. EXAMPLE
Verify that the function u(x, t) = sin(x – at)
satisfies the wave equation.
Solution Calculation gives
So u satisfies the wave equation.
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38. REVIEW
Definition of partial derivative
Notations for partial derivatives
Finding partial derivatives
Interpretations of partial derivatives
Function of more than two variables
Higher derivatives
Partial differential equations
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