finite difference Method, For Numerical analysis. working matlab code. numeric analysis finite difference method. MATLAB provides tools to solve math. Using linear programing techniques we can easily solve system of equations. This file provides a running code of Finite difference matlab code
finite difference Method, For Numerical analysis. working matlab code. numeric analysis finite difference method. MATLAB provides tools to solve math. Using linear programing techniques we can easily solve system of equations. This file provides a running code of Finite difference matlab code
Complex Analysis - Differentiability and Analyticity (Team 2) - University of...Alex Bell
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the topis is Gauss jordan and gauss elimination method.
This ppt having one example of both method and having algorithm.
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
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This ppt having one example of both method and having algorithm.
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
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MA1310 Week 6 Polar Coordinates and Complex NumbersThis lab req.docxinfantsuk
MA1310: Week 6 Polar Coordinates and Complex Numbers
This lab requires you to:
· Plot points in the polar coordinate system.
· Find multiple sets of polar coordinates for a given system.
· Convert a point from polar to rectangular coordinates.
· Convert a point from rectangular to polar coordinates.
· Plot complex numbers in the complex plane.
· Find the absolute value of a complex number.
· Write complex numbers in polar form.
· Convert a complex number from polar form to rectangular form.
· Find products of complex numbers in polar form.
· Find quotients of complex numbers in polar form.
· Find powers of complex numbers in polar form (DeMoivre's Theorem).
Answer the following questions to complete this lab:
1.
Explain why and represent the same points in polar coordinates.
2.
Match the point in polar coordinates with either A, B, C, or D on the graph.
3.
Find the rectangular coordinates of the polar point.
4. Find the polar coordinates of the rectangular point (–4, –4).
5.
Plot the complex number.
a.
b.
c.
d.
6. Find the absolute value of the complex number z = 2 + 5i.
7.
Write the complex number z = 2 – 2i in polar form. Express in degrees.
8.
Write the complex number in rectangular form.
9. Use DeMoivre's Theorem to find the indicated power of the complex number. Write answer in rectangular form.
Submission Requirements: Answer all the questions included in the lab. You can submit your answers in a Microsoft Word document, or write your answers on paper and then scan and submit the paper. Name the file as InitialName_LastName_Lab6.1_Date.
Evaluation Criteria:
· Did you show the appropriate steps to solve the given problems?
· Did you support your answers with appropriate rationale wherever applicable?
· Were the answers submitted in an organized fashion that was legible and easy to follow?
· Were the answers correct?
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Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
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Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
2. 2
INFORMATION – Wednesday 26th July 2017
•Lecture on Thursday 27 July on par. 6.5
(continue).
•No Lecture on Wednesday 9 August due to the
Public Holiday.
•Lecture on Thursday 10 August on par. 10.2.
•Class Test 1 scheduled for Thursday 3 August
on par. 6.5.
•No Tutorial on Thursday 5th October (Follows a
Monday Timetable)
4. 4
Objectives
►Definition of Polar Coordinates
►Relationship Between Polar and
Rectangular Coordinates
►Graphing Complex Numbers
►Polar Form of Complex Numbers
►De Moivre’s Theorem
►nth Roots of Complex Numbers
6. 6
Definition of Polar Coordinates
The polar coordinate system uses distances and
directions to specify the location of a point in the
plane.
To set up this system, we choose a fixed point O in
the plane called the pole (or origin) and draw from O
a ray (half-line) called the polar axis as in the figure.
7. 7
Definition of Polar Coordinates
Then each point P can be assigned polar coordinates
P(r, ) where
r is the distance from O to P
is the angle between the polar axis and the
segment
We use the convention that is positive if measured
in a counterclockwise direction from the polar axis or
negative if measured in a clockwise direction.
8. 8
Definition of Polar Coordinates
If r is negative, then P(r, ) is defined to be the point that
lies |r| units from the pole in the direction opposite to that
given by .
9. 9
Example 1 – Plotting Points in Polar Coordinates
Plot the points whose polar coordinates are given.
(a) (1, 3/4) (b) (3, –/6) (c) (3, 3) (d) ( –4, /4)
Solution:
The points are plotted.
(a) (b) (c) (d)
10. 10
Example 1 – Solution
Note that the point in part (d) lies 4 units
from the origin along the angle 5 /4,
because the given value of r is negative.
cont’d
12. 12
Relationship Between Polar and Rectangular Coordinates
The connection between the two systems is
illustrated in the figure, where the polar axis
coincides with the positive x-axis.
13. 13
Relationship Between Polar and Rectangular Coordinates
The formulas are obtained from the figure using the
definitions of the trigonometric functions and the
Pythagorean Theorem.
14. 14
Example 2 – Converting Polar Coordinates to Rectangular Coordinates
15. 15
Relationship Between Polar and Rectangular Coordinates
Note that the equations relating polar and
rectangular coordinates do not uniquely
determine r or .
When we use these equations to find the
polar coordinates of a point, we must be
careful that the values we choose for r and
give us a point in the correct quadrant.
17. 17
Graphing Complex Numbers
To graph real numbers or sets of real numbers, we
have been using the number line, which has just one
dimension.
Complex numbers, however, have two components: a
real part and an imaginary part.
This suggests that we need two axes to graph
complex numbers: one for the real part and one for
the imaginary part. We call these the real axis and
the imaginary axis, respectively.
The plane determined by these two axes is called the
complex plane.
18. 18
Graphing Complex Numbers
To graph the complex number a + bi, we plot the
ordered pair of numbers (a, b) in this plane, as
indicated.
19. 19
Example 3 – Graphing Complex Numbers
Graph the complex numbers z1 = 2 + 3i, z2 = 3 – 2i, and
z1 + z2.
Solution:
We have
z1 + z2 = (2 + 3i) + (3 – 2i)
= 5 + i.
The graph is shown.
20. 20
Graphing Complex Numbers
We define absolute value for complex numbers in a
similar fashion. Using the Pythagorean Theorem, we
can see from the figure that the distance between
a + bi and the origin in the complex plane is .
24. 24
Polar Form of Complex Numbers
Let z = a + bi be a complex number, and in the
complex plane let’s draw the line segment joining the
origin to the point a + bi .
The length of this line segment is r = | z | = .
25. 25
Polar Form of Complex Numbers
If is an angle in standard position whose
terminal side coincides with this line segment,
then by the definitions of sine and cosine
a = r cos and b = r sin
so
z = r cos + ir sin = r (cos + i sin )
30. 30
Example 5(b) – Solution
An argument is = 2/3 and r = = 2.
Thus
cont’d
31. 31
Example 5(c) – Solution
An argument is = 7/6 (or we could use = –5/6),
and r = = 8.
Thus
cont’d
32. 32
Example 5(d) – Solution
An argument is = tan–1 and r = = 5.
So
3 + 4i = 5
cont’d
33. 33
Polar Form of Complex Numbers
This theorem says:
To multiply two complex numbers, multiply the
moduli and add the arguments
To divide two complex numbers, divide the moduli and
subtract the arguments.
34. 34
Example 6 – Multiplying and Dividing Complex Numbers
Let
and
Find (a) z1z2 and (b) z1/z2.
Solution:
(a) By the Multiplication Formula
35. 35
Example 6 – Solution
To approximate the answer, we use a calculator in radian
mode and get
z1z2 10(–0.2588 + 0.9659i)
= –2.588 + 9.659i
cont’d
36. 36
Example 6 – Solution
(b) By the Division Formula
cont’d
37. 37
Example 6 – Solution
Using a calculator in radian mode, we get the
approximate answer:
cont’d
39. 39
De Moivre’s Theorem
Repeated use of the Multiplication Formula gives the
following useful formula for raising a complex number
to a power n for any positive integer n.
This theorem says:
To take the nth power of a complex number, we
take the nth power of the modulus and multiply
the argument by n.
40. 40
Example 7 – Finding a Power Using De Moivre’s Theorem
Find .
Solution:
Since , it follows from Example 5(a) that
So by De Moivre’s Theorem
43. 43
nth Roots of Complex Numbers
An nth root of a complex number z is any complex
number w such that wn = z.
De Moivre’s Theorem gives us a method for
calculating the nth roots of any complex number.
45. 45
Example 8 – Finding Roots of a Complex Number
Find the six sixth roots of z = –64, and graph these roots in
the complex plane.
Solution:
In polar form z = 64(cos + i sin ). Applying the formula
for nth roots with n = 6, we get
for k = 0, 1, 2, 3, 4, 5.
46. 46
Example 8 – Solution
Using 641/6 = 2, we find that the six sixth roots of –64 are
cont’d
48. 48
Example 8 – Solution
All these points lie on a circle of radius 2, as shown.
cont’d
The six sixth roots of z = –64
49. 49
nth Roots of Complex Numbers
When finding roots of complex numbers, we sometimes
write the argument of the complex number in degrees.
In this case the nth roots are obtained from the formula
for k = 0, 1, 2, . . . , n – 1.