diff --git a/Differentiation.md b/Differentiation.md
index dfb4595..d7a1657 100644
--- a/Differentiation.md
+++ b/Differentiation.md
@@ -1,4 +1,17 @@
-# Overview of what differentiation is:
+[<- Back to index page](https://cpawley.github.io/HHG2-MSP-Physics/)
+
+# Differentiation
+
+[ ## Overview-of-Differentiation](## Overview-of-Differentiation)
+#### Different notations
+#### Important Differentiation Rule
+#### Rules for Combined Functions
+#### Applications of Derivatives to Real Life
+#### Glossary
+#### Exercises
+#### Additional Video Links
+
+## Overview of Differentiation
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:
@@ -8,13 +21,216 @@ The next step involves understanding what the slope is, and how it can be rewrit
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?
+## Different Notations:
+
+In many study materials, you will encounter variations of the notations of **derivatives**. These different notations are denoted below, so you can recognise them.
+
+* **Leibniz’s notation:** dydxordfdx or ddxf for the first derivative of a function y or f dependent on x. The second derivatives are d2ydx2,d2fdx2, d2dx2f and respectively.
+
+* **Lagrange’s notation:** f’ for the first derivative and f’’ for the second derivative (any
+derivative beyond the third derivative - f’’’ - is denoted as fnfor the nth derivative).
+
+* **Newton’s notation:** ẏ for the first derivative and ÿ for the second.
+
+* **Euler’s notation:** Dxyor Dxf(x)for the first derivative; Dxnf(x) for the nth derivative.
+
+## Important Differentiation Rule:
+
+When a function is derived with respect to a variable, this general rule applies for finding the derivative:
+**For *f(x) = axn→ f’(x) = naxn-1* **
+
+## Rules for Combined Functions:
+
+* **Constant rule**: If a function f(x) is constant, then its derivative is zero.
+
+*f(x) = c where c is a constant, then: f’(x) = 0.*
+
+* **Sum rule:** When a function f is the summation of two functions g and h, then the derivative of f is equal to the sum of the derivatives of g and h.
+*f = αg + βh → f’ = (αg + βh)’ = αg’ + βh’.*
+
+* **Product rule:** When a function f is the product of two functions g and h, the derivative of f is the summation of the product of g and h’ and the product of g’ and h.
+*f = gh → f’ = gh’ + g’h.*
+
+* **Quotient rule:** This slightly more difficult rule is used to find the derivative of a function f that is the quotient of two functions g and h.
+*f = gh→ f’ = g'h-gh'h2*
+
+* **Chain rule:** This rule is used for composite functions, like, for example f(x) = h(g(x)). So, h is a function of g and g in turn is a function of x. The derivative of such a composite function is:
+*f’(x) = h’(g(x))g’(x).*
+
+## Applications of derivatives in real life
+
+*Business:* in business, differentiation is used to determine the profit and loss using graphs.
+
+*Optimisation:* in optimisation, differentiation is used on many occasions. Say, for example, you have a limited amount of material to produce a cylinder that holds liquid. Differentiation can help you to find out how to use the material to make a cylinder that is capable of holding the maximum amount of liquid.
+
+*Physics:* by using differentiation, a moving body’s position allows you to calculate its velocity and acceleration.
+
+*Chemistry:* in chemistry, the change in concentration of an element during a chemical reaction can be estimated using differentiation. Additionally, the rate of the reaction is found using derivatives.
+
+
+Below is a list of common derivatives.
+
+
+
+Sometimes, it can be hard to imagine how a certain graph looks yet having an image in mind may prove to be useful when working with derivatives. Below are a few graphs corresponding to some standard derivatives.
+
+
+
+
+
+**Graph 1**: y = -sin(x) which is the derivative of the cosine function. **Graph 2**: y = 1/x which is the derivative of the natural logarithm (ln).
+
+
+
+**Graph 3**: y is a cubic function, which is the derivative of a fourth-degree polynomial.
+
+
+
+Some Markdown text with some *blue* text.
+
+>span style="color:blue"> ##**Sample exercises** >/span>
+
+
+Compute the following derivatives:
+
+1. f(x) = 4x4 + 3x2 + 1
+
+2. f(x) = x sin(2x)
+
+3. f(x) = exx2
+
+4. f(x) = ln(x2)
+
+
+
+
+
+*Answers:*
+
+1. f’(x) = 16x3 + 6x
+
+2. f’(x) = sin(2x) + 2xcos(2x) (Product + Chain Rules)
+
+3. f’(x) = x2ex-2xexx4 = xex(x-2)x4= ex(x-2)x3 (Quotient rule)
+
+4. f’(x) = 1x22x =2x
+
+
+
+## Partial derivatives:
+
+Not all functions consist of only one variable; in many cases, it has multiple. This is were **partial differentiation** comes into play. Let’s use some function f(x,y,z) = 3x2 + sin(xy) + 2z which is a function of three variables: x, y, and z. **When taking a partial derivative, you differentiate the function with respect to only one of these variables; the other variables will become constants.** So, if one were to take the derivative of f(x,y,z) with respect to z, 3x2 and sin(xy) would be treated as constants. As mentioned before, the derivative of a constant is zero. Thus, the derivative will simply be 2. If one would take the derivative with respect to x, the y in sin(xy) is a constant, and the situation is similar to if you would be taking the derivative of, for example, sin(4x). The derivative of f with respect to x is then 6x + ycos(xy). Similarly, the derivative of f with respect to y is xcos(xy).
+
+The partial derivative with respect to x is denoted as f(x,y,z). So, the lowercase delta is used to denote a partial derivative.
+
+You can also take a second partial derivative, which is a partial derivative of a partial derivative. The first partial derivatives with respect to x, y, and z are respectively:
+
+> f(x,y,z) = 6x + ycos(xy)
+
+> f(x,y,z) = xcos(xy)
+
+> f(x,y,z) = 2 .
+
+No matter with respect to what variable you take a second partial derivative of z f(x,y,z), for this example function, it will always be zero. Differentiating xf(x,y,z) and yf(x,y,z) with respect to z will also result in zero, because there are only constants in these partial derivatives. xf(x,y,z) and yf(x,y,z), however, can both be differentiated with respect to x as well as y.
+
+If the former is differentiated with respect to x, that is denoted as:
+
+> 2x2f(x,y,z) .
+
+If you take the derivative with respect to y, it is denoted as:
+
+> 2ydxf(x,y,z) .
+
+> 2ydxf(x,y,z) = 6 – y2sin(xy)
+
+For 2ydxf(x,y,z), the product rule is needed.
+
+> 2ydxf(x,y,z) = cos(xy) – xysin(xy) .
+
+If you would differentiate yf(x,y,z) with respect to x, the result is most likely equal to that of
+2ydxf(x,y,z).
+
+In most cases, the order in which the two derivatives are taken have no influence on the result. However, since there are functions for which this is not the case, you should not just assume that
+
+> 2ydxf(x,y,z) = 2xdyf(x,y,z) .
+
+## Small increment approximation
+
+A useful application of (partial) differentiation is **small increment approximation.** Small changes in some function f of x and y can be approximated using the following formula:
+
+**δf = (∂ f)/(∂ x)δx + (∂ f)/(∂ y)δy**
+
+Let’s look at small increment approximation using the example of the volume of a cylinder, which is πr2h. So, the volume is dependent on two variables: the radius r and the height h. When both variables change by a very small amount, you can approximate the change in the cylinder’s volume.
+
+> δV = (∂ V)/(∂ r)δr + (∂ V)/(∂ h)δh
+
+(∂ V)/(∂ r) and (∂ V)/(∂ h) are the rates of change of V with respect to r and h, respectively; δr and δh are the changes in r and h, respectively.
+
+#### Sample Exercise:
+
+You are baking a cake that is rapidly rising in the oven. Initially, the cake had a radius of 15 cm and a height of 10 cm. At a certain point, the cake has a radius of 15.4 cm and a height of 10.12 cm. Find the approximate change in the cake’s volume.
+
+The equations:
+
+> δV = (∂ V)/(∂ r)δr + (∂ V)/(∂ h)δh
+
+> V = πr2h
+
+> (∂ V)/(∂ r) = 2πrh
+
+> (∂ V)/(∂ h) = πr2
+
+> δV = 2πrh δr + πr2 δh
+
+> δr = 4 mm
+
+> δh = 1.2 mm
+
+> δV = 2π*15 cm*10 cm*0.4 cm + π*225 cm2*0.12 cm = 462 cm3
+
+## Summary
+
+A derivative is a measurement of the rate of change with respect to a certain variable. One of the most important derivatives is as follows:
+
+For f(x) = axn→ f’(x) = naxn-1.
+
+Important differentiation rules are the product rule, the quotient rule, and the chain rule.
+
+Some functions contain multiple variables; you can take a derivative with respect to one of these variables. When differentiating with respect to one variable, the other variables in the function become constant.
+
+An important application of partial differentiation is small increment approximation, with which one can estimate small changes in a function. For some function f of x and y f(x,y), the change in f can be approximated using the following formula:
+
+> δf = (∂ f)/(∂ x)δx + (∂ f)/(∂ y)δy .
+
+
+## Glossary
+
+**Derivative:** a measurement of the **rate of change of a function with respect to a variable**. Derivatives are one of the fundamental tools for calculus.
+
+**Differentiation:** the operation of finding the derivative of a function.
+
+**Gradient/slope:** a number describing the direction and steepness of a function. For a straight line, the slope is equal to the distance travelled over the y-axis divided by the distance travelled over the x-axis. **For curved graphs, the slope at a certain point is equal to the derivative at that point.**
+
+**Tangent:** a straight line (or plane) that, at a certain point, **touches a curve but does not cross it at that point.**
+
+## Further Links to Videos and Additional Literature:
+
+
+### Videos on Rules for Combined Functions:
+
+**For an explanation and examples of the Product Rule:**
+
+[](https://www.youtube.com/watch?v=B28EXpofKy4 "For an explanation and examples of the product rule")
+
+
+**For an explanation and examples of the quotient rule:**
+
+
+[](https://www.youtube.com/watch?v=O6M4O7zY5eA "For an explanation and examples of the quotient rule)
+
+
+**For an explanation on when and how to use the chain rule:**
+
-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=H-ybCx8gt-8"For an explanation and examples of the chain rule)
-https://www.youtube.com/watch?v=EWVSxND_iWA&list=PLDesaqWTN6ESPaHy2QUKVaXNZuQNxkYQ_&index=2
diff --git a/Tensors and Contractions.md b/Tensors and Contractions.md
index fd76dac..71af963 100644
--- a/Tensors and Contractions.md
+++ b/Tensors and Contractions.md
@@ -12,7 +12,7 @@ uhh umm okay? Let us break it down!
---
-Tensor comes from the latin word “to stretch”. Tensile stress can be described as the 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!
+The word tensor comes from the latin word meaning “to stretch”. Tensile stress can be described as the stretching out of something. Tensors can measure how the (quantity sides?)/different dimensions (dimensions of what?) change (stretch) when one of the tensors 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._ (this isn't clear) The math and values might change but the physics of it will not! (what do you mean by the physics of it? the proportions/ratio of everything to each other?)
---
@@ -23,6 +23,8 @@ Tensor comes from the latin word “to stretch”. Tensile stress can be describ
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”.
+(maybe explain what indice and component and dimension mean. just a few words)
+
Below is the set up of a tensor with _m_=4 _n_=2, with _42_= 16 components:
@@ -32,7 +34,7 @@ Below is the set up of a tensor with _m_=4 _n_=2, with _42_= 16 comp
hmm still a bit abstract…
-The dimensions, _m_ is the dimension (lol). M is simply the dimension of the matrix.
+The dimension is represented by _m_ . M is simply the dimension of the matrix. (_m_ is dimension, but M is also dimension?)
3x3 matrix, _m_=3
@@ -44,27 +46,28 @@ 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, the component, within the mathematical object, matrix.
-* (matrix is a mathematical tool which can help in some scenarios organize numbers more understandable/workable).
+For rank, _n_ signifies the parameters that are necessary to find what you are looking for, which is the component, within the mathematical object, which is the matrix.
+* (a matrix is a mathematical tool which can help in to organize numbers to be more understandable and workable).
+
+Example: For a matrix which is RxR and neither R is 1, the parameter needed to find components within matrix are simply Row (look along the horizontal) and Column (look down the vertical column); therefore there are 2 parameters (“rank 2”) n=2. N is the quantity of indices, here there would be two.
-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)
+To apply this to examples, an 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!
+As we progress, and understand 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... So it's a scalar!
-n=0, 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=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
+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=3: three indices
-n=4, four indices
+n=4: four indices
and so on.
@@ -90,12 +93,13 @@ Below is a depiction of indices! There can be more than one and can be in differ
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.
+A tensor describes an object which can undergo stress. In other words: be “moved”, either stretched or compressed, along the 3 spacial directions.
+*BUT*, 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.
+Forces can not just be added along the 3-d because they are interacting differently (this is not clear, why are we talking about adding forces along the 3-d? maybe mention when you would want to do this). 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_
@@ -117,7 +121,7 @@ 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._
+_Physical nature remains the same, although the components “amounts” might change._ (what do you mean by physical nature?)
Going back to the defenition which states tensors undergo "certain transformations", below is the transformation!
@@ -130,11 +134,11 @@ Going back to the defenition which states tensors undergo "certain transformatio
-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.
+Rank 2 tensor can not be visualized like a vector or an arrow, but it can be though of as a transformation between vectors.
+Velocity is a real vector. To explain 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, for example 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]
+Angular momentum is actually a pseudo 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 pseudovector]
Ex:
@@ -148,7 +152,7 @@ The cross product with the 4-velocity will change the velocity of a charged par
# 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).
+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). (this is nicely put, can you find a way to include this at the beginning? I know it's your conclusion, but it answers some of my early confusion.)
---
---