Introduction to PDEs: Notation, Terminology and Key Concepts

Transcription

1 Chapter 1 Introduction to PDEs: Notation, Terminology and Key Concepts 1.1 Review Goal The purpose of this section is to briefly review notation as well as basic concepts from calculus. We will also introduce key concepts which will be used and studied more deeply throughout the semester Partial Derivatives Definitions Throughout these notes, we will be studying equations involving an unknown function which will be called u. u is also known as the dependent variable. u will be a function of several independent variables. How many independent variables and which variables will depend on the nature of the problem. At times, u will only depend on the space variables so that we will have u = u (x) for problems in one dimension (the real line), or u = u (x, y) for problems in two dimensions (in the plane), or u = u (x, y, z) for problems in three dimensions (in space). At other times, u will also depend on time so that u = u (x, t) or u = u (x, y, t) or u = u (x, y, z, t). x, y, and z are called the space variables and t is called the time variable. u is called the state variable because it indicates the state of the physical system. In many of the problems we will study, the space variables are usually limited to some domain we will call Ω. When Ω is bounded in at least one spatial direction, it will have a boundary we will call S. For example in the plane, that 3

2 4CHAPTER 1. INTRODUCTION TO PDES: NOTATION, TERMINOLOGY AND KEY CONCEPTS domain could be a rectangular region and the boundary would be the rectangle enclosing the region. It could also be a disk and the boundary would be the circle enclosing the disk. Definition 1 (Partial Derivative) Assuming that u = u (x, y, z, t), you will recall that the derivative of u with respect to x, denoted u, is defined to be u u (x + h, y, z, t) u (x, y, z, t) (x, y, z, t) = lim h 0 h (1.1) providing the limit exists. We will often omit (x, y, z, t) and simply write u when computing a partial derivative at a general point (x, y, z, t). We will only specify the point when it is a specific point such as u (1, 4, 2, 10). Similarly, we have the first order partial derivatives with respect to the other variables, they are u y, u u, and z t. The second order partial derivatives are the partial derivatives of these first order partial derivative. u 2 is used to denote the partial derivative with respect to x of. In other words ( ) u is denoted 2 u 2. Similarly, we have 2 u y 2, 2 u z 2, and 2 u t 2. We ( also ) have the mixed second order partial derivatives. For example, u is denoted 2 u y y while ( ) u is denoted 2 u y y. Students will recall Clairaut s theorem which states that the mixed second order partial derivatives are equal if they are continuous. Other Notations The other notations we will use are: u = u x u y = u y 2 = u xx

3 1.1. REVIEW 5 z 2 = u zz y = u yx (note the different order in which the variables appear). Students are expected to know what partial derivatives are, their various interpretations, how to compute them. Students who haven t had a multivariable class in a while may want to review this material. More specifically, students should review: 1. What are partial derivatives, what do they represent, how do we compute them. 2. Directional derivatives and gradient Gradient Definition 2 (Gradient) The gradient of a function f (x, y), denoted f (x, y) or f for short is a vector defined by f (x, y) = (f x (x, y), f y (x, y)) = f x (x, y) i + f y (x, y) j There is a smilar definition for functions of three or more variables. The gradient has many applications. We list a few here. The derivative of a function f (x, y) in the direction of a vector u at the point (a, b) is given by f (a, b) u where is the dot product between two vectors. Given a function of two variables f (x, y), f (x, y) is perpendicular to the level curves f (x, y) = k where k is a constant at any point on the level curve. Given a function of three variables F (x, y, z), F (x, y, z) is perpendicular to the level surface F (x, y, z) = k at any point on the level surface. f gives the direction of fastest increase of f Function Spaces In your calculus classes, most of the functions you worked with were continuous. You learned to differentiate and integrate functions but never really worried about their properties. This is fine for elementary mathematics. However, as you get more advanced in your studies, you will no longer be able to do that. As you use mathematical tools which are more and more advanced, you will need to be aware of the properties of the functions to which you apply those tools.

4 6CHAPTER 1. INTRODUCTION TO PDES: NOTATION, TERMINOLOGY AND KEY CONCEPTS Most tools only work on functions which have certain properties. Continuity is such a property, but there are many others. Functions are classified according to certain properties they have. By a function space, we mean a set containing functions which have certain properties. The purpose of this section is simply to name a few of these spaces we will encounter in this class. We will mention these spaces again in more details, when they are needed. An important family of function spaces involves continuity. Definition 3 (Class C k ) A function u defined on a set B is said to be of class C k on B, or we also say that u C k (B) if u and all its derivatives of order less than or equal to k are continuous. Thus, C k (B) is a set of functions, it is the set of all functions which are continuous and whose derivatives of order less than or equal to k are continuous. C 1 (B) is the set of functions defined on a set B which are continuous and whose first derivatives are also continuous. C 2 (B) is the set of functions defined on a set B which are continuous and whose first and second derivatives are also continuous. C 0 (B) is the set of functions defined on a set B which are continuous. C (B) is the set of functions defined on a set B which are continuous and whose derivatives of all orders are also continuous. Example 4 From your previous calculus classes, you already know the following: 1. If f is a polynomial in one real variable, then f C (R). 2. sin x, cos x, e x are all in C (R). There are many more function spaces. Some will be introduced later in the class Problems 1. Write the definition of u y, u u, and z t. 2. What is the geometric meaning of u, u y? 3. To evaluate 2 u, does one differentiate with respect to x first or to y? y Explain. 4. To evaluate u xy, does one differentiate with respect to x first or to y? Explain. 5. In the previous two questions, are there circumstances under which the order does not matter? (hint: you learned it in multivariable calculus).

5 1.1. REVIEW 7 6. For each function below, compute the indicated partials. (a) u (x, y) = 2x + y 2, u, u y (b) u (x, y, z) = xe 2y+z2, u, u y, u z (c) u (x, t) = sin (x + sin t), u, u t, 2 u 2, 2 u t 7. Let f (x, y) = e x2 +y 2. Find f and find the derivative of f in the direction of u = (1, 1) at the point (1, 2). 8. Let f (x) = x. (a) Find the largest set B on which we can say f C 0 (B). (b) Find the largest set B on which we can say f C 1 (B). 9. Let B be some set and let m and n be two nonnegative integers. Are there circumstances under which C m (B) C n (B)? If yes, what are they?