Hi! My name is **Alex**, and I've just graduated from the University of Cambridge with a first-class Master's degree in Astrophysics. I'm about to begin a PhD in Physics at King's College London, where I will also be working as a Graduate Teaching Scholar, helping to teach university-level physics. **I love teaching** and learning about maths and science, and I have **lots of experience** sharing my enthusiasm: from tutoring IB students and training successful maths competition teams, to demonstrating experiments at children's science fairs.

Hi! My name is **Alex**, and I've just graduated from the University of Cambridge with a first-class Master's degree in Astrophysics. I'm about to begin a PhD in Physics at King's College London, where I will also be working as a Graduate Teaching Scholar, helping to teach university-level physics. **I love teaching** and learning about maths and science, and I have **lots of experience** sharing my enthusiasm: from tutoring IB students and training successful maths competition teams, to demonstrating experiments at children's science fairs.

I will do everything I can to make sure our sessions are **productive, engaging, fun, and tailored to you**. We will cover the material that you need help with, in a way that suits **you**, at the pace that **you** set. I will focus on explaining concepts **in a way that makes sense to you**, using analogies and ideas that you are comfortable with.

If you have any questions at all, please get in touch with me, or click to book a free "Meet the Tutor" session. I'm looking forwards to hearing from you!

I will do everything I can to make sure our sessions are **productive, engaging, fun, and tailored to you**. We will cover the material that you need help with, in a way that suits **you**, at the pace that **you** set. I will focus on explaining concepts **in a way that makes sense to you**, using analogies and ideas that you are comfortable with.

If you have any questions at all, please get in touch with me, or click to book a free "Meet the Tutor" session. I'm looking forwards to hearing from you!

Enhanced DBS Check

11/01/20175from 2 customer reviews

Samia (Student)

March 17 2016

Alexander has a very good knowledge of the subject (physics) and explained the questions very well and was prepared for the session.

Maria (Student)

January 17 2016

Alex is amazing at explaining difficult concepts - I took two sessions for IB Physics before the mock examinations and as a result, I moved my prediction up by a a grade, thank you!

"Integration by parts" is one of several methods in our arsenal that we can use to integrate a function *f*(*x*). It involves expressing *f*(*x*) as the product of two other functions, which I will call *u*(*x*) and *v'*(*x*) (where *v*' = ^{dv}/_{dx} is the first derivative of *v* with respect to *x*):

∫ *f*(*x*) d*x = *∫ *u*(*x*) · *v*'(*x*) d*x* = *u*(*x*) · *v*(*x*) - ∫ *u*'(*x*) · *v*(*x*) d*x*

This formula comes from the product rule for differentiation. If we differentiate the product *u*(*x*) · *v*(*x*) with respect to *x* using the product rule, we get the following:

^{d}/_{dx}(*u*(*x*) · *v*(*x*)) = *u*(*x*) · *v*'(*x*) + *u*'(*x*) · *v*(*x*)

Integrating both sides of this expression with respect to *x* removes the ^{d}/_{dx }on the left hand side, and creates two integrals on the right hand side:

*u*(*x*) · *v*(*x*) = ∫ *u*(*x*) · *v*'(*x*) d*x + *∫ *u*'(*x*) · *v*(*x*) d*x*

If we subtract the final term from both sides, we arrive at our original formula again:

∫ *f*(*x*) d*x = *∫ *u*(*x*) · *v*'(*x*) d*x* = *u*(*x*) · *v*(*x*) - ∫ *u*'(*x*) · *v*(*x*) d*x*

But why would we want to use this method? Well, integration by parts is generally useful in cases such as the one below:

∫ Ln(*x*)/*x*^{2} d*x*

At a first glance, we have no idea how to integrate this function. We do, however, know how to integrate 1/*x*^{2 }(which is our *v*' in this case), and we know how to differentiate Ln(*x*) (which is our *u*). So, using integration by parts:

∫ Ln(*x*)/*x*^{2} d*x = *∫ Ln(*x*) · (1/*x*^{2}) d*x = *(-1/*x*) · Ln(*x*) - ∫ (-1/*x*) · (1/*x*) d*x *

*= *-Ln(*x*)/*x* + ∫ (1/*x*^{2}) d*x = *-Ln(*x*)/*x* - 1/*x + C*

(Not forgetting our constant of integration at the end!)

"Integration by parts" is one of several methods in our arsenal that we can use to integrate a function *f*(*x*). It involves expressing *f*(*x*) as the product of two other functions, which I will call *u*(*x*) and *v'*(*x*) (where *v*' = ^{dv}/_{dx} is the first derivative of *v* with respect to *x*):

∫ *f*(*x*) d*x = *∫ *u*(*x*) · *v*'(*x*) d*x* = *u*(*x*) · *v*(*x*) - ∫ *u*'(*x*) · *v*(*x*) d*x*

This formula comes from the product rule for differentiation. If we differentiate the product *u*(*x*) · *v*(*x*) with respect to *x* using the product rule, we get the following:

^{d}/_{dx}(*u*(*x*) · *v*(*x*)) = *u*(*x*) · *v*'(*x*) + *u*'(*x*) · *v*(*x*)

Integrating both sides of this expression with respect to *x* removes the ^{d}/_{dx }on the left hand side, and creates two integrals on the right hand side:

*u*(*x*) · *v*(*x*) = ∫ *u*(*x*) · *v*'(*x*) d*x + *∫ *u*'(*x*) · *v*(*x*) d*x*

If we subtract the final term from both sides, we arrive at our original formula again:

∫ *f*(*x*) d*x = *∫ *u*(*x*) · *v*'(*x*) d*x* = *u*(*x*) · *v*(*x*) - ∫ *u*'(*x*) · *v*(*x*) d*x*

But why would we want to use this method? Well, integration by parts is generally useful in cases such as the one below:

∫ Ln(*x*)/*x*^{2} d*x*

At a first glance, we have no idea how to integrate this function. We do, however, know how to integrate 1/*x*^{2 }(which is our *v*' in this case), and we know how to differentiate Ln(*x*) (which is our *u*). So, using integration by parts:

∫ Ln(*x*)/*x*^{2} d*x = *∫ Ln(*x*) · (1/*x*^{2}) d*x = *(-1/*x*) · Ln(*x*) - ∫ (-1/*x*) · (1/*x*) d*x *

*= *-Ln(*x*)/*x* + ∫ (1/*x*^{2}) d*x = *-Ln(*x*)/*x* - 1/*x + C*

(Not forgetting our constant of integration at the end!)

**Olbers' paradox** is a thought experiment that arises in cosmology* which shows that our universe **does not fit **the "Static Newtonian" model.

Sir Isaac Newton came up with a model for the universe which can be described by **four basic statements**:

i) The universe is **infinitely large**

ii) The universe is **infinitely old**

iii) The universe is **not expanding or contracting** (i.e. it is "static")

iv) All the planets, stars, galaxies etc. are **roughly evenly distributed** throughout the universe (i.e. it has "large-scale homogeneity")

This came to be referred to as the **Static Newtonian** model.

Olbers' paradox is a logical argument which attempts to show that our universe **cannot fit this model.** It does this by arguing the following:

If the universe was infinitely large, and had a fairly even distribution of stars throughout it, then this would mean that there would be an **infinite number of stars** surrounding the Earth on all sides. Furthermore, if this universe was infinitely old, then the light from all of those stars would have had **enough time to reach the Earth.** With an infinite amount of stars on all sides, constantly emitting an infinite amount of light, our sky would always be **infinitely bright** - there would be no difference between day and night. **This is obviously not the case!**

We can see that our four statements have lead us to a model that **does not agree **with our observation that the sky is dark at night. Therefore **at least one**, or **possibly several** of these statements must be false. This leads us to the following **conclusions**: either,

a) the universe is **finite in size**, and thus has a finite number of stars

b) the universe has some **finite age**, and thus the light from most stars has not had enough time to reach us

c) the stars, planets etc. are **not evenly spread** on a large scale, which would also allow for a finite number of stars

d) some combination of a), b) and c).

These conclusions lead us to consider **more sophisticated** models of the universe, such as the **Big Bang model**, which states that a) and b) are true, but not c): i.e. the universe is **expanding** (and therefore **not static**, unlike Newton's model).

*The study of our universe; its structure, how it changes with time, how it began, etc.

**Olbers' paradox** is a thought experiment that arises in cosmology* which shows that our universe **does not fit **the "Static Newtonian" model.

Sir Isaac Newton came up with a model for the universe which can be described by **four basic statements**:

i) The universe is **infinitely large**

ii) The universe is **infinitely old**

iii) The universe is **not expanding or contracting** (i.e. it is "static")

iv) All the planets, stars, galaxies etc. are **roughly evenly distributed** throughout the universe (i.e. it has "large-scale homogeneity")

This came to be referred to as the **Static Newtonian** model.

Olbers' paradox is a logical argument which attempts to show that our universe **cannot fit this model.** It does this by arguing the following:

If the universe was infinitely large, and had a fairly even distribution of stars throughout it, then this would mean that there would be an **infinite number of stars** surrounding the Earth on all sides. Furthermore, if this universe was infinitely old, then the light from all of those stars would have had **enough time to reach the Earth.** With an infinite amount of stars on all sides, constantly emitting an infinite amount of light, our sky would always be **infinitely bright** - there would be no difference between day and night. **This is obviously not the case!**

We can see that our four statements have lead us to a model that **does not agree **with our observation that the sky is dark at night. Therefore **at least one**, or **possibly several** of these statements must be false. This leads us to the following **conclusions**: either,

a) the universe is **finite in size**, and thus has a finite number of stars

b) the universe has some **finite age**, and thus the light from most stars has not had enough time to reach us

c) the stars, planets etc. are **not evenly spread** on a large scale, which would also allow for a finite number of stars

d) some combination of a), b) and c).

These conclusions lead us to consider **more sophisticated** models of the universe, such as the **Big Bang model**, which states that a) and b) are true, but not c): i.e. the universe is **expanding** (and therefore **not static**, unlike Newton's model).

*The study of our universe; its structure, how it changes with time, how it began, etc.