1.1.2 - Example 2
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Introduction to PDEs
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Today, we're going to learn about Partial Differential Equations, or PDEs for short. Can anyone tell me what a PDE is?
Isn’t it an equation that involves partial derivatives of functions with multiple variables?
Exactly! PDEs are fundamental in describing systems in fields like physics and engineering. They are crucial for modeling phenomena such as heat transfer and fluid dynamics. Let's dive deeper into how we can form these equations.
How do we actually form a PDE from an equation?
Great question! We can form a PDE by eliminating arbitrary constants or functions through differentiation. Let's start with the constants.
Eliminating Arbitrary Constants
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Let's look at an equation: z = ax + by + c, where a, b, and c are constants. Who can tell me the first step to eliminate these constants?
We need to partially differentiate with respect to x and y, right?
Correct! When we differentiate, we find that ∂z/∂x = a and ∂z/∂y = b. We can then set p and q to these values. What do you think the next step is?
We substitute p and q back into the original equation?
Exactly! By substituting, we can eliminate c and express our PDE. In this case, we arrive at a relationship that leads to a PDE. It’s essential to practice this method!
Eliminating Arbitrary Functions
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Now, let's consider when arbitrary functions are involved. For example, if z = f(x² + y²), how do we differentiate?
We should use the chain rule for differentiation.
Correct! When using the chain rule, we differentiate to find p = 2x f'(u) and q = 2y f'(u). Now, how can we eliminate f'?
We can relate p and q to find expressions for f' and eliminate it?
Exactly! This method is key to forming a PDE in cases involving arbitrary functions.
Summary of Key Concepts
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Let's summarize what we've learned today. Can anyone tell me the two methods we discussed for forming PDEs?
We can eliminate arbitrary constants and arbitrary functions!
Correct! Each method requires different approaches for differentiation and elimination. Consistent practice will help us master this skill.
Are there any real-world applications of these methods?
Absolutely! These formations are essential in mathematical modeling and solving real-life problems in engineering and physics.
Introduction & Overview
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Quick Overview
Standard
The section elaborates on the formation of PDEs, demonstrating methods to eliminate arbitrary constants and functions through differentiation. Examples illustrate the practicality of these methods in generating PDEs relevant to real-world applications.
Detailed
Formation of Partial Differential Equations (PDEs)
Partial Differential Equations (PDEs) incorporate partial derivatives of a function with multiple independent variables and arise in various scientific fields such as fluid dynamics and electromagnetism. The formation of a PDE involves eliminating arbitrary constants or functions from an equation to derive a more general equation that characterizes an entire class of functions meeting specific conditions. This section details two primary methods of forming PDEs:
- Eliminating Arbitrary Constants: Through partial differentiation with respect to independent variables, arbitrary constants can be systematically removed from the equation.
- Example: For the equation z = ax + by + c, differentiation leads to expressions for partial derivatives, from which you can eliminate constants and form a PDE.
- Eliminating Arbitrary Functions: In cases where arbitrary functions are present, a similar differentiation approach applies, but it often requires additional steps to establish relationships between the derivatives.
- Example: In z = f(x² + y²), partial differentiation allows us to relate these variables to form the necessary PDE.
Understanding these methods is crucial as they enable us to represent complex physical scenarios mathematically.
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Introduction to Example 2
Chapter 1 of 5
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Chapter Content
Given:
z=a²+x²+y²
Detailed Explanation
In Example 2, we start with a function z represented by the equation z = a² + x² + y². Here, a is a constant and x, y are independent variables. This equation shows how z depends on both x and y, with a² as an arbitrary constant that needs to be eliminated to form a partial differential equation (PDE). The goal is to derive a PDE involving the partial derivatives associated with the independent variables.
Examples & Analogies
Think of z as the height of a hill (z) at any point on a 2D plane represented by x (east-west) and y (north-south) coordinates. The constant a² represents a baseline height that is fixed, similar to how the sea level is constant while the hill's height varies.
Differentiation Step
Chapter 2 of 5
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Chapter Content
Differentiate:
∂z/∂x = 2x, ∂z/∂y = 2y
Detailed Explanation
The next step is to differentiate z partially with respect to x and y. This involves using the rules for differentiation. As a result, we find the first partial derivative of z with respect to x is 2x, and with respect to y, it's 2y. These derivatives (2x and 2y) help us understand how z changes as we change x and y, respectively.
Examples & Analogies
Imagine you are walking on the hill mentioned earlier. As you go east (x-direction), the height increase (z) per step you take can be thought of as the derivative ∂z/∂x. The derivative tells you how steep the hill is at point x.
Substituting Variables
Chapter 3 of 5
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Let:
p=2x,
q=2y
⇒ x= p/2, y= q/2
Detailed Explanation
In this step, we introduce new variables p and q to represent the derivatives we just calculated. We let p = 2x and q = 2y. This allows us to express x and y in terms of p and q, which simplifies our process of eliminating the constants. Thus, we write x as p/2 and y as q/2.
Examples & Analogies
Continuing with the hill analogy, think of p and q as indicators of how much further you move in the east (x) and north (y) directions at a certain point on the hill. By organizing these movements into their own quantities (p and q), we simplify our calculations.
Substituting into the Original Equation
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Chapter Content
Substitute into original:
z=a²+x²+y² = a² + (p/2)² + (q/2)²
⇒a² = z - (p² + q²)/4
Detailed Explanation
Next, we substitute our expressions for x and y back into the original equation for z. This results in rewriting z as: z = a² + (p/2)² + (q/2)². By rearranging the equation, we isolate a², resulting in a² = z - (p² + q²)/4. This manipulation is crucial because it brings us closer to eliminating the arbitrary constant a².
Examples & Analogies
Imagine we're trying to find out how high the hill (z) is without considering the fixed base height (a²). By adjusting p and q to account for how far you've stepped (thus changing x and y), we can express the hill's height in a more flexible way that doesn't depend on the fixed base.
Forming the PDE
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Chapter Content
Thus, the required PDE is:
p² + q² = 4z
Detailed Explanation
Finally, we eliminate the constant a², leading us to the partial differential equation p² + q² = 4z. This equation expresses the relationship between the derivatives (p and q) and the variable z without the arbitrary constant, thus forming a proper PDE for this system.
Examples & Analogies
In our hill example, we now have a relationship that tells us how the steepness (p and q) relates to the height (z) over a portion of the hill. We've eliminated the constant baseline (a²), focusing solely on how shifts in terrain affect one another.
Key Concepts
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Formation of PDEs: The process of deriving PDEs involves eliminating arbitrary constants or functions through differentiation.
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Partial Differentiation: A technique used to find the derivative of a function with respect to one variable, holding others constant.
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Chain Rule: A mathematical rule used to differentiate compositions of functions, particularly useful when dealing with arbitrary functions.
Examples & Applications
Given z = ax + by + c, differentiate to eliminate constants to form a proper PDE.
For z = f(x² + y²), use the chain rule to express the relationship and develop the PDE.
Memory Aids
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Rhymes
To form a PDE, don’t be shy, differentiate first, give constants a bye!
Stories
Imagine a scientist trying to understand a fluid's behavior. They write down an equation, but it contains extra constants. After taking a deeper dive, they differentiate and eliminate those constants, revealing the PDE that governs fluid flow.
Memory Tools
C-FED: Clear Function, Eliminate Derivatives to form PDEs - remember to differentiate first!
Acronyms
PDE
Partial Derivatives Emerge - helpful to remember that partial derivatives are the foundation of PDEs.
Flash Cards
Glossary
- Partial Differential Equation (PDE)
An equation that contains the partial derivatives of a multivariable function.
- Arbitrary Constants
Constants in an equation that can take on any value, allowing flexibility in the function's shape.
- Arbitrary Functions
Functions that are not defined within the context of the original equation and can represent a wide range of behaviors.
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