Practice Spherical (9.3) - Partial Differential Equations - Mathematics III (PDE, Probability & Statistics)

Practice: Spherical

Lesson 9.3Partial Differential Equations

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9 questions

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2 cards from this section. Quick, good the night before a test.
Exercise
6 questions · about 4 min · pass mark 70%. Hints available.
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Quiz
3 questions · about 2 min · pass mark 70%. Hints available.
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6 sample questions

Question 1.
What is the general formula for the Laplacian in spherical coordinates?
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Make sure to incorporate radius and angle derivatives in your answer.
Question 2.
Can the Laplacian be applicable to a function only in terms of $r$ ?
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Consider what happens to the angles if they are constant.
3Question 3.
What is the purpose of the Laplacian operator?
- A measure of curved space.
- It measures the average behavior of a function.
- Calculating volume of solid figures.
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Think about properties of functions.
4Question 4.
True or False: The Laplacian can only be applied to two-dimensional functions.
- True
- False
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Consider the dimensionality of coordinate systems.
Question 5.
Given a function $u(r, \theta, \phi) = r^3 \sin(\theta)$ , calculate its Laplacian. Show the step-by-step derivation.
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Remember, you need to differentiate and apply product rules.
Question 6.
Design a spherical model to analyze heat dissipation in a metal sphere. Derive the differential equations involved.
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Boundary conditions are critical; think of insulated versus exposed edges.

5 more questions in this section. Enroll free to practise them all.