In this video I go over another example on integrating rational functions by the method of partial fractions and this time solve the integral of 1/(x^2 - a^2) where a is not equal to 0. The result of this integral is a useful formula which can be used to simplify and solve integrals where a polynomial in the denominator can be written as a difference of squares.
In this video I model the Earth as an ellipsoid as used by the World Geodetic System of 1984 (WGS-84) since it is a more accurate model than a sphere. The Earth is not perfectly spherical but instead is more squished vertically by about 21.6 km: the distance from the center to the Poles is 6356.523 km while the distance to the equator is larger at 6378.137 km. The parallels or circular lines of equal latitude are the horizontal circular traces of the ellipsoid. The meridians or elliptical lines of equal longitude are the vertical elliptical traces of the ellipsoid. I plot this all out using GeoGebra's 3D and 2D graphing calculators (the 2D traces turn into cylinders in 3D so I had to also use the 2D graphing calculator).
GeoGebra 3D graphing calculator: https://www.geogebra.org/3d/tabys4ec
GeoGebra 2D graphing calculator: https://www.geogebra.org/calculator/hte4yr46
Time stamps:
- Exercise 1: Modeling Earth as an Ellipsoid: 0:00
- Solution to (a): Earth's Ellipsoid Equation: 1:09
- Solution to (b): Parallels or Circular Lines of Equal Latitude: 6:26
- Note: Using GeoGebra's 2D (not 3D) calculator to plot 2D circles: 10:37
- Solution to (c): Meridians or Elliptical Lines of Equal Longitude: 11:58
Full video below:
- Cylinders and Quadric Surfaces: https://youtu.be/k4EyLr4uOUA
- HIVE video notes: https://peakd.com/hive-128780/@mes/cylinders-and-quadric-surfaces
- Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EfmYwT81SebJnU1Ny8Fc5u
- Infinite Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ .
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In this video I go over what indefinite integrals are and show how they are a really convenient way of writing and tabulating anti-derivatives. I also show some very important notes that you should always keep in mind while working with indefinite integrals, so this video is a must watch!
Download the notes in my video: https://www.dropbox.com/s/97a7ilf277h264i/235%20-%20Indefinite%20Integrals.pdf
Related Videos:
Fundamental Theorem of Calculus - Introduction and Part 1 of the Theorem: http://youtu.be/3o8Q6UJzJyk
Fundamental Theorem of Calculus - Intro and Proof of Part 2 of the Theorem: http://youtu.be/yuIl-BPQHss
Antiderivatives - A brief Introduction: http://youtu.be/zDzUHqAUSEw
The Definite Integral - Brief Introduction: http://youtu.be/vhMP5SKbQjU
Derivative of Trigonometry Functions: Derivative of tan(x): http://youtu.be/Jf6p4crjrSc
Continuity and Open and Closed Intervals: http://youtu.be/u29fd3WYT-g
Derivative of Power Functions Part 1 - x^n where n is a positive integer: http://youtu.be/-Yv85MZNYgU
Definition of Derivative Simple Explanation: http://youtu.be/0rjGMpM06Eg .
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https://www.youtube.com/watch?v=_jl9Zy2sBwk
In #MESExperiments 37 I demonstrate some very interesting observations of permanent magnets and unmagnetized steel when placed on a large magnet. Small magnets tend to move to the center of the magnet but when lifted they suddenly move to the edge. Small unmagnetized steel balls and screws tend to move to the edge of the magnet both when placed and lifted. However, a larger magnetic ball and large unmagnetized steel bolt both tend to move to the center both when placed and lifted. I also show a cylinder magnet wants to stand vertically in the center but moves to edge when lifted. When placed flat and pushed towards the center, it can launch away!
Finally I show that a tiny magnet behaves completely opposite of a tiny unmagnetized steel screw, both in position and angle direction. The tiny screw balances near the center at and an angle pointing away from center and gradually flattens out when moved to the edge. The tiny magnet balances near the edge at an angle towards the center and gradually flattens out when moved to the center. These are all some very interesting and puzzling experiments which I hope will help us unravel the mystery that is electromagnetism!
Original unedited experiment: https://youtu.be/3kcJ2XSzbe0
Timestamps:
- Unmagnetized steel vs magnets: 0:00
- Magnetic ball rolls to center of magnet: 0:05
- Magnetic ball moves to edge when lifted: 0:11
- Unmagnetized steel ball moves to edge of magnet: 0:32
- Steel ball is unstable in center: 0:44
- Steel ball moves to edge when lifted: 0:55
- Large magnetic ball moves to center of magnet: 1:20
- Large magnetic balls moves to center even when lifted: 1:31
- Large unmagnetized steel bolt moves to center of magnet even when lifted: 1:50
- Additional Experiments: 2:15
- Cylinder magnet moves to center vertically: 2:20
- Cylinder magnet moves to edge when lifted: 2:27
- Pushing cylinder magnet away causes it to flip back to center: 2:32
- When flat, cylinder magnet keeps poles aligned to center and edge: 2:36
- When flat, cylinder magnet launches away when pushed past center: 2:45
- Small unmagnetized steel screw moves to edge of magnet: 3:00
- Small screw stands at an angle near center: 3:09
- Small screw moves to edge when lifted: 3:14
- Tiny unmagnetized steel screw stands at an angle near center: 3:21
- Tiny screw moves to edge when lifted: 3:30
- Tiny screw becomes more vertical closer to center: 3:36
- Angle of screw is away from center: 3:44
- Tiny magnet balances at an angle near edge of larger magnet: 3:58
- Angle of tiny magnet is towards the center and flattens out near center: 4:03
- Tiny magnet moves to and along edge: 4:11
- Tiny screw and magnet balance in opposite positions and angles: 4:27
- Tiny screw flattens out at edge, Tiny magnet flattens out at center: 4:40
Stay tuned for #MESExperiments 38...
Experiment screenshots and playlists:
- Screenshots of the Experiment: https://peakd.com/hive-128780/@mes/kkgywcda
- #MESExperiments video series: https://www.youtube.com/playlist?list=PLai3U8-WIK0HUHoank-Lj9q6RGAS51QRh
- Hive playlist: https://peakd.com/mesexperiments/@mes/list
- DRAFT #MESExperiments video series: https://mes.fm/experiments-draft
Related Videos:
#MESScience video series: https://www.youtube.com/playlist?list=PLai3U8-WIK0GhjCHmTw1XbqMD_EdVKdd9
#AntiGravity video series: https://peakd.com/antigravity/@mes/series
#FreeEnergy video series: https://www.youtube.com/playlist?list=PLai3U8-WIK0FKVpHL_onhaqBeVP-8qJXV .
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In this video I quickly recap on absolutely and conditionally convergent series. An absolutely convergent series is such that the absolute value of all the terms of a series is convergent. Since the absolute value of all the terms means that the series sums up to its maximum size, if it is absolutely convergent then it means it is also convergent. On the other hand, if a series is convergent but NOT absolutely convergent, then it is said to be conditionally convergent. This arises from positive and negative terms canceling out.
The timestamps of key parts of the video are listed below:
- Question 6: 0:00
- (a) Absolutely convergent series: 0:14
- (b) Absolutely convergent series are convergent: 0:56
- (c) Conditionally convergent series: 1:12
This video was taken from my earlier video listed below:
- Infinite Sequences and Series: Review and True-False Quiz: https://youtu.be/F0dsQLdXXpI
- HIVE video notes: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-review-and-true-false-quiz
- Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FCqXVJv1r7eJvrvphfkr6L
Related Videos:
Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ .
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...
https://www.youtube.com/watch?v=xL8C5edXS-A
Adam Eisenberg, a direct witness at the Pentagon, and Norman aka 9/11 Revisionist join the show to discuss the Pentagon attack on 9/11 and what actually happened.
September 14, 2024 SATURDAY at 1 PM PST / 3 PM CST / 4 PM EST / 10 PM CAT / 21:00 GMT (UK)
Guest links:
- Adam's Instagram: https://www.instagram.com/awakenedadam
- Norman's Substack: https://substack.com/@911revision
Stream notes and playlist:
- Stream notes and links: https://peakd.com/hive-113182/@mes/livestream-54-pentagon-911
- MES Livestreams: https://www.youtube.com/@mes/streams
- Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FqwyUa_ICwTlqO0S6Y3kAn .
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This stream is created with #PRISMLiveStudio
In this video I go over some more useful examples on the chain rule for derivatives and look at functions within functions within functions and thus require using the chain rule multiple times.
Download the notes in my video: https://www.dropbox.com/s/uqlpxfn5gyxeq59/414%20-%20Chain%20Rule%20Examples%20Part%204.pdf
Related Videos:
Chain Rule Examples: Part 1: http://youtu.be/j1YKstIMIFc
Chain Rule Examples: Part 2: http://youtu.be/kX6ourBH9KE
Chain Rule Derivatives: Examples Part 3: http://youtu.be/J769tWFpy70
Derivative Rules: Proof of Chain Rule: http://youtu.be/tYDDpKzP-VU
Derivative Examples using the Chain Rule: http://youtu.be/KB9_lBxTrwI
Derivative of Trigonometry Functions: Derivative of sin(x): http://youtu.be/elEvQ4Wu7Pk
Derivative of Trigonometry Functions: Derivative of cos(x): http://youtu.be/LMjVp-GsrCw
Derivative of Trigonometry Functions: Derivative of tan(x): http://youtu.be/Jf6p4crjrSc
Trigonometry: Derivative of sec(x): Proof: http://youtu.be/0sO0TehqMOw
The Number e - A Brief Introduction and it's Derivative: http://youtu.be/o_s_YYD6v3g .
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https://www.youtube.com/watch?v=3neWNOz1Wf8
In this video I go over another example on predator-prey systems for population growth of two species, but this time modify the Lotka-Volterra equations to instead model two species that compete or cooperate in obtaining resources. Two different sets of differential equations are presented, and in this example I show the steps involved in determining what kind of model each set is describing. This is done by seeing how the population growth of one species is affected by the increases or decrease of the population of the other species. If it's a mutual increase in population growth whenever either population increases, then this represents a cooperation model, such as a bees or insects pollinating flowers. On the other hand, if the populations of either species decreases with the increase of the other species, then this represents a model that shows two species fighting for the same species, such as a pair of plant-eating animals. This is a very good video to show how to read and understand systems of differential equations and how they can be used to model various real world phenomenon.
Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIht4Lx_Db7-g03Axxpg
View Video Notes on Steemit: https://steemit.com/mathematics/@mes/predator-prey-systems-example-3
Related Videos:
Predator-Prey Systems: Example 2: Part 2: https://youtu.be/ngfH2uoaWGg
Predator-Prey Systems: Example 2: Part 1: https://youtu.be/LD2-ARezF7E
Predator-Prey Systems: Example 1: Part 2: https://youtu.be/Jvpl1qpBLa8
Predator-Prey Systems: Example 1: Part 1: https://youtu.be/dkQYqI4FXxA
Differential Equations: Predator-Prey Systems: https://youtu.be/b3NCjsZhQdQ
Linear Differential Equations: https://youtu.be/T3MliJDBaxc
Differential Equations: Exponential Growth and Decay: https://youtu.be/DZtDUIZuxcg
Differential Equations: Separable Equations: https://youtu.be/pBV-xT9ty94
Differential Equations: Population Growth: Logistic Equation: https://youtu.be/yE8aoY8Bks4 .
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https://www.youtube.com/watch?v=y_Kv97u9Q50
In this video I go over the same example as in the last video surface area but this time write the resulting surface area integral in terms of y and dy as opposed to x and dx. This is possible because the arc length can be written in either notation and thus the resulting integral and final answer is the same.
Download the notes in my video: https://onedrive.live.com/redir?resid=88862EF47BCAF6CD!103729&authkey=!AAhYJ6Jugs1t1yk&ithint=file%2cpdf
View Video Notes on Steemit: https://steemit.com/mathematics/@mes/applications-of-integrals-surface-area-example-2-solution-2
Related Videos:
Applications of Integrals: Surface Area: Example 2: Solution 1: https://youtu.be/5nUzb02pDcI
Applications of Integrals: Surface Area: Example 1: https://youtu.be/E9rwIVUifuA
Applications of Integrals: Surface Area: https://youtu.be/JkDPmAD37qk
Applications of Integrals: Arc Length Proof: https://youtu.be/2rb4H_rmgxg
Applications of Integrals: Arc Length Function: https://youtu.be/MWKK3qLvSwU
Derivative of y = x^n - Part 2: General Power Rule: http://youtu.be/Ibvu33oh49o .
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https://www.youtube.com/watch?v=nlL8ePAYT50
I use the definition of the Derivative to derive the derivative of exponential functions of the form f(x) = a^x. I also allude to the number e which is a very important number and greatly simplifies the derivative of exponential functions.
Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIioI5ltm9PU86OymIUw?e=cZxrwD
View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/derivative-of-ax-exponential-functions-part-1-using-definition-of-derivative
Related Videos:
Definition of Derivative Simple Explanation: http://youtu.be/0rjGMpM06Eg
Power Functions Part 2: What is x^0 and 0^0???: http://youtu.be/VzFUDiLzRiE
Exponential Functions and Graphing : http://youtu.be/FFNwh8KR9-Q .
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