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zU7nwJzM=BDVE|%eoA=3c3J*REPXu5TIpSVuVfe!rLFkWCxC}*Iek=0OqO7k5tN>CI zYukE!lqieR_>SwE{X^9b>fHekMpc07TeOg8Y;1~ZYDFg-M3Ri&ii)uCaE$sZIt+YT zJ6l^7Y>^;&U&!$#9T9_W#aotDTO9(--C`DmsomwpR?dg6XV}Uuz)?$MVx3G(fi%?C z_Qq)^Ia!2D=Q52&;jPC5bs%UBkc5GFI#x?jVk^}eIB+F{H-H-8e_EclVUPLXa7Gk1zd#hCr-N>56eT$|NTrNP@c#i9Y*h*X literal 0 HcmV?d00001 From 3f813a23eb0e65bdfd6de38ea6957bf885c41b4b Mon Sep 17 00:00:00 2001 From: sadVries <60645668+sadVries@users.noreply.github.com> Date: Thu, 25 Jun 2020 16:03:44 +0200 Subject: [PATCH 02/19] Delete Coordinate-Systems.md --- Coordinate-Systems.md | 29 ----------------------------- 1 file changed, 29 deletions(-) delete mode 100644 Coordinate-Systems.md diff --git a/Coordinate-Systems.md b/Coordinate-Systems.md deleted file mode 100644 index 77b6d83..0000000 --- a/Coordinate-Systems.md +++ /dev/null @@ -1,29 +0,0 @@ - -# Coordinate Systems -Coordinate systems are systems for visually representing information along numerical axes, which are called dimensions. Enabling us for the first time to represent algebraic equations graphically as shapes – such as lines, curves, parabolas, or circles – giving these equations a geometric form for the first time. -## Cartesian Coordinates -The Cartesian or the fundamental coordinate system is a powerful tool to accurately define position and thus help other quantities which rely on accurate initial measurements. So, we can use coordinate systems to accurately visualize and understand what function are and how they behave. -### 1-D Coordinate system -Plotting on 1 dimension means only plotting on one axis, creating a number line. A number line assigns an origin (0) for reference and then positive and negative number on either side. An example of its use is plotting a motion for example a piston moving up and down. - -#### 2-D Coordinate system -A 2-dimensional plot adds one axis to our 1-dimensional coordinate system. Now we label de axis and call the horizontal axis x and the vertical y. A 2-s coordinate system has its origin where the x and y axis intersect and this is called the origin. The axis basically split up the coordinate system into 4 quadrants named quadrant + a roman numeral accordingly. To define a point on the coordinate system we use the format (x, y). So point a on figure a is (4, 5) with 4 being the value on the x-axis and 5 on the y-axis. 2d coordinate system has a multitude of uses like plotting movement on a terrain from a birds-eye view. - -### 3-D Coordinate system -To get a 3-d coordinate system you add another axis to your x- and y-axis called the z-axis and thus adding a 3rd dimension. With 3-axis we can measure any type movement in space because are system now has all 6 ranges of freedom that an object can have. - - - - -## Spherical coordinate system -The spherical coordinate system describes a point in three-dimensional space. The position is described by three numbers: radial distance (r), polar angle (θ) and azimuthal angle (φ). The spherical coordinate system is mostly used for solving problems that deal with spherical symmetry like gravity of a Planet. - -- r, the radial distance, radius or radial coordinate is the distance from the origin to the point. -- θ is the angle counter-clockwise from polar or positive x-axis in the xy-plane. The polar angle is also called colatitude, zenith angle, normal angle, or inclination angle -- φ is the angle between the positive z-axis and the point - -what the spherical coordinate system does is map a point on a sphere with radius r. then determine with angle θ how far along a circle it lies that is part of a sphere and then determine the height of that circle with angle φ. The link below is a live model in which you can tweak r, θ, φ in order to change the point on the sphere.
-https://www.wolframcloud.com/objects/demonstrations/SphericalCoordinates-source.nb -## Coordinate Conversions -For simple point of coordinates the conversion between the coordinate systems are give by table 1 - From 1d84e95f38267be81c07671d3c84f2807056ff0d Mon Sep 17 00:00:00 2001 From: sadVries <60645668+sadVries@users.noreply.github.com> Date: Fri, 26 Jun 2020 11:22:14 +0200 Subject: [PATCH 03/19] Add files via upload --- Conversion_table.png | Bin 0 -> 56705 bytes Del_on_coordinate_systems.png | Bin 0 -> 34812 bytes Unit_vector_conversion.png | Bin 0 -> 62861 bytes 3 files changed, 0 insertions(+), 0 deletions(-) create mode 100644 Conversion_table.png create mode 100644 Del_on_coordinate_systems.png create mode 100644 Unit_vector_conversion.png diff --git a/Conversion_table.png b/Conversion_table.png new file mode 100644 index 0000000000000000000000000000000000000000..fdd3985cf423293ccd8e8514ec509b1b10315866 GIT binary patch literal 56705 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Date: Sat, 27 Jun 2020 16:11:09 +0200 Subject: [PATCH 12/19] Delete Differentiation.md --- Differentiation.md | 20 -------------------- 1 file changed, 20 deletions(-) delete mode 100644 Differentiation.md diff --git a/Differentiation.md b/Differentiation.md deleted file mode 100644 index dfb4595..0000000 --- a/Differentiation.md +++ /dev/null @@ -1,20 +0,0 @@ -# Overview of what differentiation is: - -Differentiation is, by definition, the way to find a function that represents the rate of change of one variable with respect to another variable. To begin simply, we will take a variable that changes across time, like a car traveling along a road for a defined amount of time. By differentiating this, we can find the velocity. By further differentiating that, the acceleration can be found. Why this is will be further discussed below: - -The primary motivation for understanding and using differentiation was the tangent line problem, that is, for a curved graph, what is the slope that is tangent to it at a given point. Thus, by solving the slope of the curve at that point, the tangent problem can be solved. In many cases finding the slope may be easy. If a graph has the equation y=mx+b, m already represents the slope. It becomes more difficult if your graph follows an equation similar to y=cos(x)+x3at x=2. - -The next step involves understanding what the slope is, and how it can be rewritten to something that looks like a differentiation. The slope/gradient of a curve is commonly expressed as “rise over run” where over a straight line finite line, the slope can be expressed as the total distance it travels over the y-axis (the rise) divided by the total distance it travels with respect to the x-axis (the run). In other words, it is the change of variables within the y-axis divided by the change in variables in the x-axis. This change is known as delta (Δ). Going back to the example of the car going down a road, if it travels 30 metres in 5 seconds, and we take the distance to be the y-axis and the time to be the x-axis, then the slope represents 305m/sor 6m/s; the velocity. - -It won't always be as easy as this though, as most graphs in physics are curved, which makes getting the slope much harder to get. In this way, a tangential line is used to represent the slope at a given point. The definition of a tangent is something that only just touches the line, such as in the image below. The point that it intersects at is infinitesimally small, and will from now on be referred to as “d” instead of delta (which represents a finite difference). Therefor, the slope at a tangential point on the graph of position versus time will be written as dxdt; a differentiation. - -Some good websites to read about the laws of differentiation, and practice are as follows: -Differentiation and Integration - Academic Skills Kit - (ASK)https://brilliant.org/wiki/derivative-by-first-principle/ -https://en.wikibooks.org/wiki/Calculus/Differentiation/Differentiation_Defined#What_is_Differentiation? - -Or if video is preferred, these are also useful tools: - -https://www.youtube.com/watch?v=-_POEWfygmU&list=PL96AE8D9C68FEB902https://www.youtube.com/watch?v=-_POEWfygmU&list=PL96AE8D9C68FEB902 - -https://www.youtube.com/watch?v=EWVSxND_iWA&list=PLDesaqWTN6ESPaHy2QUKVaXNZuQNxkYQ_&index=2 From 3620c50c81fa5b546256451d3930650459aa7431 Mon Sep 17 00:00:00 2001 From: sadVries <60645668+sadVries@users.noreply.github.com> Date: Sat, 27 Jun 2020 16:12:21 +0200 Subject: [PATCH 13/19] Delete Dot Cross Product.md --- Dot Cross Product.md | 47 -------------------------------------------- 1 file changed, 47 deletions(-) delete mode 100644 Dot Cross Product.md diff --git a/Dot Cross Product.md b/Dot Cross Product.md deleted file mode 100644 index cbc9da7..0000000 --- a/Dot Cross Product.md +++ /dev/null @@ -1,47 +0,0 @@ -The two most important vector operations are the dot- and cross product. Both of these are based on matrix operations, and while the dot product is easy to do in vector notation, the cross product might be easier to do in matrix notation. - -# Dot Product - -The dot product is similar to multiplying vectors, creating a scalar number. When the vectors are not split up into their components, the result of the dot product is - -![alt text](https://github.com/cpawley/HHG2-MSP-Physics/blob/Floris-Images/1%20(2).png), - -where θ is the angle between the two vectors. When the components of the vectors are known, meaning there is an equation like Ax + By + Cz, the dot product is performed like this - -![alt text](https://github.com/cpawley/HHG2-MSP-Physics/blob/Floris-Images/2.png) - -As shown, the like parts are multiplied together, and the products are added together. The dot product is both cumulative and distributive, meaning the order of the vectors does not matter, and A ・ (B + C) is the same as A・B + A・C. -Due to the angle in the operation, the dot product is the product of the two vectors if they are parallel, and 0 if they are perpendicular. - -# Cross Product - -The cross product is completely different from the dot product, especially when it comes to performing it with the components. In normal vector form, the cross product equals - -![alt text](https://github.com/cpawley/HHG2-MSP-Physics/blob/Floris-Images/3.png) - -The n with the hat in this equation is a unit vector in the direction perpendicular to the plane of A and B. This means that if A is in the x direction, and B in the y, the cross product will be a new vector in the z direction. When the vectors’ components are known, the cross product is performed by - -![alt text](https://github.com/cpawley/HHG2-MSP-Physics/blob/Floris-Images/4.png) - -Due to the disordered nature of this formula, the cross product often can be referred from a 3x3 matrix, like this one. - -![alt text](https://github.com/cpawley/HHG2-MSP-Physics/blob/Floris-Images/5%20(2).png) - -The cross product for the x component can then be calculated by taking the determinant of the 2x2 matrix created by the y and z components. This is done by multiplying the cross terms (Ay and Bz, and Az and By), and subtracting the second term from the first, as seen in the original formula for the cross product with components. -Although the cross product is distributive, the order of the vectors does matter, meaning it is not associative. -Like in the dot product, the cross product is 0 or AB depending on the angle. However, since there is a sin in front of the angle, the cross product is 0 when the vectors are parallel, and equal to AB when they are perpendicular. - -For some extra explanations, mostly on properties, and practice, you can visit: - -https://tutorial.math.lamar.edu/Classes/CalcII/CrossProduct.aspx - -https://tutorial.math.lamar.edu/Classes/CalcII/DotProduct.aspx - -For helpful videos, you can scroll through these videos from Khan Academy - -https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/dot-cross-products/v/vector-dot-product-and-vector-length - -Formulas from Griffiths, D. J. (2005). Introduction to electrodynamics. - - - From 55677ac2fe54228dd6cb3fdaa28cd2d06571b484 Mon Sep 17 00:00:00 2001 From: sadVries <60645668+sadVries@users.noreply.github.com> Date: Sat, 27 Jun 2020 16:12:33 +0200 Subject: [PATCH 14/19] Delete Fourier Transform.md --- Fourier Transform.md | 25 ------------------------- 1 file changed, 25 deletions(-) delete mode 100644 Fourier Transform.md diff --git a/Fourier Transform.md b/Fourier Transform.md deleted file mode 100644 index e6c91c3..0000000 --- a/Fourier Transform.md +++ /dev/null @@ -1,25 +0,0 @@ -Almost everything in our world can be described as a wave function, and any wave function is a sum of sinusoids of different frequencies. This means that the wave function can also be represented as a set of frequencies. From this, the original wave function is able to be changed by filtering out certain unwanted frequencies or boosting a frequency. The image below shows how a single sinusoidal wave looks as a frequency. Since a function can be graphed as a bunch of sinusoidal waves, a Fourier Transform graph with peaks at the waves' frequencies can be created. - - -![alt text](https://github.com/cpawley/HHG2-MSP-Physics/blob/Floris-Images/6.png) - -(http://mri-q.com/fourier-transform-ft.html) - -In electronics, the original data might not be a wave function, but it can be represented as one. This wave function can then get filtered by a combination of resistors and capacitors. The order and values of these determines what type of filter it will be, and what frequencies it will filter out. This can for example take any noise out of the data, or only select a certain range of important frequencies. One will still have the same data in the original way, for example a voice file, but it will sound better due to the filtering. - - -Some websites with extra information: - -• http://www.thefouriertransform.com/ - -• https://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/ - -• http://rundle.physics.ucdavis.edu/PHYGEO30/Fourier_Transforms.pdf - -Some videos if those are preferred: -• https://www.youtube.com/watch?v=spUNpyF58BY - -For explanations on the more mathematical Fourier Series: -https://www.youtube.com/watch?v=r6sGWTCMz2k&t=8s, also the videos linked at the end of this one - - From 03e1cbd09f9c43ef72324e836968412c18b86316 Mon Sep 17 00:00:00 2001 From: sadVries <60645668+sadVries@users.noreply.github.com> Date: Sat, 27 Jun 2020 16:12:47 +0200 Subject: [PATCH 15/19] Delete Tensors and Contractions.md --- Tensors and Contractions.md | 131 ------------------------------------ 1 file changed, 131 deletions(-) delete mode 100644 Tensors and Contractions.md diff --git a/Tensors and Contractions.md b/Tensors and Contractions.md deleted file mode 100644 index 0344853..0000000 --- a/Tensors and Contractions.md +++ /dev/null @@ -1,131 +0,0 @@ - -# Tensors and Contractions - ---- - -## Tensors: - -“A tensor is a mathematical object which transforms as a tensor” - ---- -uhh umm okay? Let us break it down! - ---- - -Tensor comes from the latin word “to stretch”. Tensile stress can be described as the feeling of stretching something out. This is what tensors can measure, the quantity sides/different dimensions change (stretch) when one of them is changed. A tensor has this neat ability to remain a tensor and have the same physics occur when is it transformed – when one changes its coordinate system. The math and values might change but the physics of it will not! - - ---- -### okay but how is this measured and calculated? - ---- - -First, perhaps a different definition: -A rank – _n_ tensor in _m_ – dimensions is a mathematical object that has: _n_ indices and _mn_ components and obeys “certain transformation rules”. - -Below is the set up of a tensor with _m_=4 _n_=2: - -![alt text](https://github.com/cpawley/HHG2-MSP-Physics/blob/Floris-Images/stresstensor.png) - - ---- - -hmm still a bit abstract… - -The dimensions, _m_ is the dimension (lol). M is simply the dimension of the matrix. - -3x3, _m_=3 - -4x4, _m_=4 - -The amount of components within a tensor is _mn_. What is _n_? - ---- - -the infamous _n_ is... “rank” - -For rank, n signifies the parameters that are necessary to find what you are looking for within the mathematical object. It is the numerical amount of information needed to find selection (component) within a matrix (matrix is a mathematical tool which can help in some scenarios organize numbers more understandable/workable). -Example: For a matrix which is RxR and neither R is 1, the parameter needed to find components within matrix are simply Row and Column (look at along row and down column) therefore there are 2 parameters (“rank 2”)  n=2. N is the quantity of indices, here there would be two. -If the matrix was Rx1, then n=1, because we would only need to look from one direction. While if the matrix changed to an IxJxK (a 3-d matrix) then there would be 3 parameters, n=3, three directions to search needed to find desired component. -For example, the electromagnetic tensor is a rank n=2 and a dimension m=4 tensor (4x4). (insert picture) - -As we progress, and understands tensors better, we begin to use index notation. -When the tensor symbol (usually a Greek letter) does not have an index it means it is rank 0, which means one doesn’t need any information to find a component – there is only one value attached to tensor….. it is a scalar! - -n=0, scalar. - -n=1, one piece of info, one index, necessary to find component, vector. Example velocity of a rolling ball on surface. - -n=2, two pieces of information, two indices - -Traditionally for 2- and 3-dimensions Latin letters are used. 4 dimensions and higher Greek letters are used. - -n=3, three indices - -n=4, four indices - -and so on. - ---- -### a bit better?? - ---- - -Well now more on the physics of a tensor! - -A tensor can describe an object which can undergo stress, be “moved”, either stretched or compressed, along the 3 spacial directions. *BUT*, then also each side of the object can also undergo structural strain (6 more stresses) – therefore it can be said there are 9 possible stresses. - -![alt text](https://github.com/cpawley/HHG2-MSP-Physics/blob/Floris-Images/9s.png) - - -Forces can not just be added along the 3-d because they are interacting differently. Each of the 9 forces makes the object “react in a different way”. A way to express this tensor is with a 3x3 matrix, m = 3. - -So then how many components total? _mn_ - -if there is a 4x4 matrix, _m_=4 -and _n_=2 - -_mn_ = _42_ = 16 - -It is the same as saying 4x4 = 16.. but if its a rank _n_=3, then there would be 3-d matrix (again m=4): 4x4x4 = _43_ = 64 - - ---- -### Okay, okay I think I get it... but - ---- - -What makes them tensors? - -* HOW THEY TRANSFORM (COORDINATE SYSTEM TRANSFORMATION) - * chosen coordinate system should and does not affect the physics of the situation - -_Physical nature remains the same, although the components “amounts” might change._ - -Rank 2 tensor can not be visualized like an vector, arrow, but it can be though of as a transformation between vectors. -Velocity is a real vector. If a vector is zero in one coordinate system, for it to be a real vector, it should be zero in all coordinate systems. -Velocity can be relative, the ball moves relative to the table but not relative to itself. - -angular momentum is actually a psudo vector, because if the origin changes, the time can go to zero, therefore changing the actual physics not just the maths of this vector) [magnetic field is also a psudovector] - - -Ex: - -F=q(v x B+E) - - -![alt text](https://github.com/cpawley/HHG2-MSP-Physics/blob/Floris-Images/fqbv.png) - -The cross product with the 4-velocity will change the velocity of a charged particle into a force. - - -# Tensors CONCULSION -It is a number or a collection of numbers which maintain its MEANING under coordinate transformations. The components of the tensor will change, but the physical nature will not (after coordinate transformation). - ---- ---- - - - - - From c9127bfb0d918c21213d402338b92914f0e476e4 Mon Sep 17 00:00:00 2001 From: sadVries <60645668+sadVries@users.noreply.github.com> Date: Sat, 27 Jun 2020 16:12:58 +0200 Subject: [PATCH 16/19] Delete Lagrangian.md --- Lagrangian.md | 66 --------------------------------------------------- 1 file changed, 66 deletions(-) delete mode 100644 Lagrangian.md diff --git a/Lagrangian.md b/Lagrangian.md deleted file mode 100644 index c9578f8..0000000 --- a/Lagrangian.md +++ /dev/null @@ -1,66 +0,0 @@ -# **Lagrangian Mechanics** - -Newton's laws of motion are the foundation on which all of classical mechanics, from celestial mechanics to rotational motion, to the ideal gas law, is based. However, identifying all the forces between objects creates difficulties when applying the Newtonian algorithm. -While vector quantities, in which Newton’s laws are described, can be rewritten in any coordinate system to simplify this problem, it is not the ideal option. - -This calls for a different coordinate system to simplify calculations. - - -It would be useful if the important quantities could be easily rewritten in the more optimal coordinate system and then solved. This can be done by describing our systems in terms of scalars instead of vectors. Scalars are quantities, such as mass or energy, that are described by a number or magnitude alone – and therefore are the same in any coordinate system. This concept is known as Langrangian (mechanics). - - -To clarify why this system is helpful, take a brief look at some challenging problems for the Newtonian approach: - -### **Mass on a sliding wedge** - -Imagine we have a block-shaped mass on a sliding wedge. Both the block and the wedge are free to move without friction under the force of gravity. -![Image of mass on a sliding wedge](https://ds055uzetaobb.cloudfront.net/brioche/uploads/3Q6ItWBg5y-screen-shot-2015-08-21-at-84207-pm.png?width=2000) - -Setting this problem up is difficult for two reasons. -1. The mass MM moves along the floor while mm moves at an angle, which creates a tension in the choice of coordinates (the Cartesian grid in the frame of the ground, and another in the frame of the tilted surface). -2. The second is that the normal force between MM and mm depends upon the motion of MM and mm. - -### **Bead on a spinning hoop** - -Consider a bead that is free to slide on a rotating hoop. This is difficult to describe in vectors because gravity calls for Cartesian coordinates, but the constraint of the hoop suggests polar coordinates. Deciding on a uncomplicated representation is difficult. - -![Image of a bead on a spinning hoop](https://ds055uzetaobb.cloudfront.net/brioche/uploads/Jg7mhai6ay-screen-shot-2015-08-21-at-92015-pm.png?width=600) - -### **Coupled pendulums** - -Lastly, the coupled pendulum: where one pendulum hangs from the end of another. Between the coupling of the two pendulums, and the constraint that keeps the two pendulums from breaking apart, the choice of representation is not straightforward. - - - -The standard of identifying forces, writing down constraints, converting to a useful system of coordinates, and substituting to uncouple equations does not work easily here. - - -## Problems with the Newtonian approach - -To see why Langrangian is needed, we must explore the problems with Newtonian mechanics. - -- **Vector representation**: We run into trouble using vectors when we need to switch coordinate systems. - - Though we can write the force of gravity as *-gẑ* in a Cartesian coordinate system, it’s easier to describe the pendulum swing in terms of the angle *θ*. Rewriting the acceleration vector of the pendulum bob is unnecessarily difficult. It only gets worse in more complex problems. -Contrast this to the ease with which we can transform scalar quantities such as the kinetic energy: - -![img](http://www.sciweavers.org/download/Tex2Img_1593075746.jpg) - - - -A system without vectors would be useful. - -- __Identifying forces__: In using Newton's laws, we must account for all forces as well as their counter forces. When there is more than one particle, this can lead to many incompatibly described forces. Life, or at least complex systems, would be easier without vector forces. - -- __Constraints__: Newtonian mechanics needs explicit constraints in the equations of motion, which can quickly become very complicated, especially when constraints are imposed by non-rigid surfaces. Langrangian imposes constraint implicitly by the choice of coordinates. - - - -We aim to reshape mechanics in an energy-based, vector-free manner. -## How? - -Newton's second law states that the acceleration of any particle is given by the net force upon the particle, where the acceleration and net force are both vectors in one, two, or three dimensions: - -We recall the work-energy principle which states that the work done by the net force (i.e. the non-constraint forces) on a particle is equal to the increase in kinetic energy of the particle. We ignore constraint forces because they always act perpendicular to the direction of motion, and thus perform no work on systems. - -:thinking: - From 6cad75f911fcc38b206c9b1545aff5a04a28ed5a Mon Sep 17 00:00:00 2001 From: sadVries <60645668+sadVries@users.noreply.github.com> Date: Sat, 27 Jun 2020 16:13:06 +0200 Subject: [PATCH 17/19] Delete Tensors.md --- Tensors.md | 112 ----------------------------------------------------- 1 file changed, 112 deletions(-) delete mode 100644 Tensors.md diff --git a/Tensors.md b/Tensors.md deleted file mode 100644 index 5078f1e..0000000 --- a/Tensors.md +++ /dev/null @@ -1,112 +0,0 @@ - -# Tensors and Contractions - ---- - -## Tensors: - -“A tensor is a mathematical object which transforms as a tensor” - ---- -uhh umm okay? Let us break it down! - ---- - -Tensor comes from the latin word “to stretch”. Tensile stress can be described as the feeling of stretching something out. This is what tensors can measure, the quantity sides/different dimensions change (stretch) when one of them is changed. A tensor has this neat ability to remain a tensor and have the same physics occur when is it transformed – when one changes its coordinate system. The math and values might change but the physics of it will not! - ---- -### okay but how is this measured and calculated? - ---- - -First, perhaps a different definition: -A rank – _n_ tensor in _m_ – dimensions is a mathematical object that has: _n_ indices and _mn_ components and obeys “certain transformation rules”. - ---- - -hmm still a bit abstract… - -The dimensions, _m_ is the dimension (lol). M is simply the dimension of the matrix. - -3x3, _m_=3 - -4x4, _m_=4 - -The amount of components within a tensor is _mn_. What is _n_? - ---- - -the infamous _n_ is... “rank” - -For rank, n signifies the parameters that are necessary to find what you are looking for within the mathematical object. It is the numerical amount of information needed to find selection (component) within a matrix (matrix is a mathematical tool which can help in some scenarios organize numbers more understandable/workable). -Example: For a matrix which is RxR and neither R is 1, the parameter needed to find components within matrix are simply Row and Column (look at along row and down column) therefore there are 2 parameters (“rank 2”)  n=2. N is the quantity of indices, here there would be two. -If the matrix was Rx1, then n=1, because we would only need to look from one direction. While if the matrix changed to an IxJxK (a 3-d matrix) then there would be 3 parameters, n=3, three directions to search needed to find desired component. -For example, the electromagnetic tensor is a rank n=2 and a dimension m=4 tensor (4x4). (insert picture) - -As we progress, and understands tensors better, we begin to use index notation. -When the tensor symbol (usually a Greek letter) does not have an index it means it is rank 0, which means one doesn’t need any information to find a component – there is only one value attached to tensor….. it is a scalar! - -n=0, scalar. - -n=1, one piece of info, one index, necessary to find component, vector. Example velocity of a rolling ball on surface. - -n=2, two pieces of information, two indices - -Traditionally for 2- and 3-dimensions Latin letters are used. 4 dimensions and higher Greek letters are used. - -n=3, three indices - -n=4, four indices - -and so on. - ---- -### a bit better?? - ---- - -Well now more on the physics of a tensor! - -A tensor can describe an object which can undergo stress, be “moved”, either stretched or compressed, along the 3 spacial directions. *BUT*, then also each side of the object can also undergo structural strain (6 more stresses) – therefore it can be said there are 9 possible stresses. -Forces can not just be added along the 3-d because they are interacting differently. Each of the 9 forces makes the object “react in a different way”. A way to express this tensor is with a 3x3 matrix, m = 3. - -So then how many components total? _mn_ - -if there is a 4x4 matrix, _m_=4 -and _n_=2 - -_mn_ = _42_ = 16 - -It is the same as saying 4x4 = 16.. but if its a rank _n_=3, then there would be 3-d matrix (again m=4): 4x4x4 = _43_ = 64 - - ---- -### Okay, okay I think I get it... but - ---- - -What makes them tensors? - -* HOW THEY TRANSFORM (COORDINATE SYSTEM TRANSFORMATION) - * chosen coordinate system should and does not affect the physics of the situation - -_Physical nature remains the same, although the components “amounts” might change._ - -Rank 2 tensor can not be visualized like an vector, arrow, but it can be though of as a transformation between vectors. -Velocity is a real vector. If a vector is zero in one coordinate system, for it to be a real vector, it should be zero in all coordinate systems. -Velocity can be relative, the ball moves relative to the table but not relative to itself. - -angular momentum is actually a psudo vector, because if the origin changes, the time can go to zero, therefore changing the actual physics not just the maths of this vector) [magnetic field is also a psudovector] - - -Ex: - -F=q(v x B+E) - - (insert matrix) - -The cross product with the 4-velocity will change the velocity of a charged particle into a force. - - -# CONCULSION -It is a number or a collection of numbers which maintain its MEANING under coordinate transformations. The components of the tensor will change, but the physical nature will not (after coordinate transformation). From 67472446767b079aeba5264f018b14e7e9f7a7c8 Mon Sep 17 00:00:00 2001 From: sadVries <60645668+sadVries@users.noreply.github.com> Date: Sat, 27 Jun 2020 16:13:15 +0200 Subject: [PATCH 18/19] Delete index.md --- index.md | 37 ------------------------------------- 1 file changed, 37 deletions(-) delete mode 100644 index.md diff --git a/index.md b/index.md deleted file mode 100644 index a5d44b9..0000000 --- a/index.md +++ /dev/null @@ -1,37 +0,0 @@ -## Welcome to GitHub Pages - -You can use the [editor on GitHub](https://github.com/cpawley/HHG2-MSP-Physics/edit/master/index.md) to maintain and preview the content for your website in Markdown files. - -Whenever you commit to this repository, GitHub Pages will run [Jekyll](https://jekyllrb.com/) to rebuild the pages in your site, from the content in your Markdown files. - -### Markdown - -Markdown is a lightweight and easy-to-use syntax for styling your writing. It includes conventions for - -```markdown -Syntax highlighted code block - -# Header 1 -## Header 2 -### Header 3 - -- Bulleted -- List - -1. Numbered -2. List - -**Bold** and _Italic_ and `Code` text - -[Link](url) and ![Image](src) -``` - -For more details see [GitHub Flavored Markdown](https://guides.github.com/features/mastering-markdown/). - -### Jekyll Themes - -Your Pages site will use the layout and styles from the Jekyll theme you have selected in your [repository settings](https://github.com/cpawley/HHG2-MSP-Physics/settings). The name of this theme is saved in the Jekyll `_config.yml` configuration file. - -### Support or Contact - -Having trouble with Pages? Check out our [documentation](https://help.github.com/categories/github-pages-basics/) or [contact support](https://github.com/contact) and we’ll help you sort it out. From df4b577ae4606c5e145dbe21bd10782dd3e26ddf Mon Sep 17 00:00:00 2001 From: sadVries <60645668+sadVries@users.noreply.github.com> Date: Sat, 27 Jun 2020 19:16:51 +0200 Subject: [PATCH 19/19] Delete _config.yml --- _config.yml | 1 - 1 file changed, 1 deletion(-) delete mode 100644 _config.yml diff --git a/_config.yml b/_config.yml deleted file mode 100644 index c741881..0000000 --- a/_config.yml +++ /dev/null @@ -1 +0,0 @@ -theme: jekyll-theme-slate \ No newline at end of file