Podcast
Questions and Answers
Which activity typically involves a longer rental period?
Which activity typically involves a longer rental period?
- Hiring skis for a day trip.
- Renting a flat for a year. (correct)
- Hiring a boat for an afternoon.
- Renting a bicycle for a weekend.
Which of the following phrases best describes a town that is full of people and has a lot of noise?
Which of the following phrases best describes a town that is full of people and has a lot of noise?
- Friendly and helpful.
- Noisy and crowded. (correct)
- Beautiful and lovely.
- Warm and sunny.
If someone says their holiday was 'awful,' which of the following words would have a similar meaning?
If someone says their holiday was 'awful,' which of the following words would have a similar meaning?
- Wonderful
- Fantastic
- Lovely
- Horrible (correct)
Which of the following is the best way to describe the activity shown in picture 5?
Which of the following is the best way to describe the activity shown in picture 5?
If you plan to 'go by bus', which other options align with this form of transportation?
If you plan to 'go by bus', which other options align with this form of transportation?
Which activity from the list involves exploring historical or interesting places?
Which activity from the list involves exploring historical or interesting places?
Which set of adjectives would you likely use to describe a luxurious resort?
Which set of adjectives would you likely use to describe a luxurious resort?
Which of these activities might you do if you 'go swimming'?
Which of these activities might you do if you 'go swimming'?
What kind of weather is best described as 'very windy, foggy, cloudy'?
What kind of weather is best described as 'very windy, foggy, cloudy'?
Which of the following activities is most likely represented by picture 8, showing a girl in a tent?
Which of the following activities is most likely represented by picture 8, showing a girl in a tent?
When is it more appropriate to use 'rent' instead of 'hire'?
When is it more appropriate to use 'rent' instead of 'hire'?
If a traveler wants to 'take photos', what other action might they also do?
If a traveler wants to 'take photos', what other action might they also do?
In the context of holidays, which activity is best represented by picture 6, showing people on skis?
In the context of holidays, which activity is best represented by picture 6, showing people on skis?
Which phrase describes a holiday that includes going to different attractions and landmarks?
Which phrase describes a holiday that includes going to different attractions and landmarks?
What do 'great', 'lovely' and 'wonderful' have in common?
What do 'great', 'lovely' and 'wonderful' have in common?
Which adjectives would best describe a town that is both beautiful to look at and friendly to visitors?
Which adjectives would best describe a town that is both beautiful to look at and friendly to visitors?
According to the image, what date in Brighton is circled on the calendar?
According to the image, what date in Brighton is circled on the calendar?
What phrases contain the word 'go'?
What phrases contain the word 'go'?
Which activity could involve 'walking' or 'cycling'?
Which activity could involve 'walking' or 'cycling'?
Which of the following actions is the most sensible way to spend your time?
Which of the following actions is the most sensible way to spend your time?
Flashcards
Go Abroad
Go Abroad
To go to a foreign country.
Go Away for the Weekend
Go Away for the Weekend
To leave for a short trip, usually Friday to Sunday.
Go By Bus/Car/Plane/Train
Go By Bus/Car/Plane/Train
To travel using public transport or a vehicle.
Go Camping
Go Camping
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Go for a Walk
Go for a Walk
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Go on Holiday
Go on Holiday
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Go Out at Night
Go Out at Night
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Go Sightseeing
Go Sightseeing
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Go Skiing
Go Skiing
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Go Swimming
Go Swimming
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Sunbathe
Sunbathe
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Stay in a hotel
Stay in a hotel
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Book a flight online
Book a flight online
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Buy Souvenirs
Buy Souvenirs
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Take photos
Take photos
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Friendly, Helpful
Friendly, Helpful
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Beautiful, Lovely
Beautiful, Lovely
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Warm, Sunny
Warm, Sunny
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Awful, Horrible
Awful, Horrible
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Windy, Foggy, Cloudy
Windy, Foggy, Cloudy
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Study Notes
- A partial differential equation (PDE) involves an unknown function and its partial derivatives.
- PDEs model phenomena like heat transfer, fluid flow, wave propagation, and electromagnetism.
General Form of a PDE
- The general form of a PDE is F(x, y, u, ux, uy, uxx, uyy, uxy,...) = 0.
- x, y are independent variables.
- u = u(x, y) is the dependent variable.
- ux, uy are the first-order partial derivatives.
- uxx, uyy, uxy are the second-order partial derivatives.
Examples of PDEs
- Heat Equation: ∂u/∂t = α (∂²u/∂x²) describes temperature changes over time.
- Wave Equation: ∂²u/∂t² = c² (∂²u/∂x²) describes wave propagation.
- Laplace's Equation: (∂²u/∂x²) + (∂²u/∂y²) = 0 describes steady-state phenomena.
Order and Linearity
- Order: The highest order derivative in the equation.
- Heat, wave, and Laplace equations are second-order PDEs.
- Linearity: The unknown function and its derivatives appear linearly.
- Heat, wave, and Laplace equations are linear PDEs.
Methods for Solving PDEs
- Analytical solutions for PDEs are limited.
- Numerical methods approximate solutions; examples include:
- Finite Difference Method
- Finite Element Method
- Finite Volume Method
Boundary Conditions
- Boundary conditions specify values of the unknown function on the domain's boundary.
- Dirichlet: u(x, y) = f(x, y) on the boundary.
- Neumann: ∂u/∂n = g(x, y) on the boundary where n is the normal direction.
- Robin: αu + β(∂u/∂n) = h(x, y) on the boundary.
Applications
- PDEs are used across various fields:
- Physics: Electromagnetism, fluid dynamics, quantum mechanics
- Engineering: Structural analysis, heat transfer, control systems
- Finance: Option pricing, risk management
- Biology: Population dynamics, disease modeling
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