Notes on multivariable calculus

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1 Notes on multivariable calculus Jonathan Wise February 2, Review of trigonometry Trigonometry is essentially the study of the relationship between polar coordinates and Cartesian coordinates in the plane See the next section for more details about this The unit circle is the collection of all points in the plane that are a distance 1 from the origin 1

2 If θ is an angle, cos(θ) and sin(θ) are the Cartesian coordinates of the point on the unit circle making an angle θ with the positive x-axis Here are some of the most important properties of sin and cos Since the points (sin(θ), cos(θ)) is a distance 1 from the origin, we have sin(θ) 2 + cos(θ) 2 = 1 ( ) by the Pythagorean theorem These are the two most important properties in trigonometry, so I will repeat them 1 A point on the unit circle making an angle θ with the positive x-axis has Cartesian coordinates (cos(θ), sin(θ)) 2 For any number θ, the sum cos(θ) 2 + sin(θ) 2 is always 1 Most other important properties of trigonometric functions can be deduced from these For example, suppose we divide the equality ( ) by cos(θ) 2 Then we get sin(θ) 2 cos(θ) = 1 cos(θ) 2 If we remember that the definition of tan(θ) is sin(θ) cos(θ) and the definition of sec(θ) is 1 cos(θ) then this equality becomes tan(θ) = sec(θ) 2 ( ) The moral is that as long as you can remember ( ), there is no need to memorize ( ) because you can rederive it easily from ( ) 2 Two- and three-dimensional space 21 Coordinate systems A coordinate system is a language for describing points in a space 211 Cartesian coordinates in 2 dimensions The Cartesian coordinate system may be the most straightforward one In 2-dimensional Cartesian coordinates, we agree on a point to call the origin and two perpendicular axes, usually called the X- and Y -axes To describe a point P in Cartesian coordinates, we give directions for getting from the origin to P by moving parallel to the axes The Cartesian coordinates of P can be written as an ordered pair (x, y) One arrives at P by starting at the origin and travelling x units parallel to the X-axis and then y units parallel to the Y -axis If x or y is negative then one should travel backwards by the absolute value of x or y when moving parallel to the corresponding axis Example 1 If P = ( 5, 3) in Cartesian coordinates then P is in the upper left quadrant, 5 units to the left of the y-axis and 3 units above the x-axis 212 Polar coordinates Another system that is useful in situations with circular symmetry is called the polar coordinate system In polar coordinates, we choose an origin again and just one axis (which we ll call the X-axis) We describe a point P by saying how far it is from the origin and how large an angle it makes with the positive X-axis If P has polar coordinates (r, θ) then to arrive at P, starting at the origin, we travel r units along the X-axis and then rotate by an angle θ around the origin 2

3 Remark 1 Note that the notation P = (u, v) could now mean two completely different things, depending on whether we are working in Cartesian or polar coordinates We must therefore be careful to specify which coordinate system we are using when we write that sort of thing If it is not clear from context which coordinate system we are working in, it s a good idea to specify it in the notation For example, one could write P = (u, v) polar to emphasize that (u, v) are polar coordinates of P and not the Cartesian coordinates (or some other kind of coordinates) Remark 2 Note that, contrary to the situation in Cartesian coordinates, it makes a difference which of the two directions we follow first If we were to rotate around the origin before moving along the X-axis, we would always find ourselves on the X-axis in the end! This is one of the key advantages of the Cartesian system: we can follow the directions in any order we please Example 2 A single point can have more than one representation in polar coordinates! For example, the coordinates (3, π 4 ), the coordinates ( 3, 5π 4 ) and the coordinates (3, 9π 4 ) all describe the same point We can translate from polar coordinates to Cartesian coordinates using trigonometry If (x, y) are the Cartesian coordinates of a point P and (r, θ) are polar coordinates of the same point, then x = r cos(θ) y = r sin(θ) It is trickier to give a rule for finding polar coordinates for a point from Cartesian coordinates This is because there can be more than one way to represent a single point in polar coordinates, as we saw in Remark 1 In fact, it is impossible to give a single, continuous rule for translating from Cartesian to polar coordinates The best we can do is to give several rules that apply at different times For example, if P is to the right of the Y -axis and has Cartesian coordinates (x, y) (thus x > 0), we then one way to write P in polar coordinates is 213 Cartesian coordinates in 3 dimensions r = x 2 + y 2 y θ = arctan x Cartesian coordinates in 3 dimensions are similar to those in 2 dimensions, except there is now a third axis perpendicular to both of the other two and called the Z-axis A point is written with three coordinates: P = (x, y, z) 214 Cylindrical coordinates Cylindrical coordinates are a sort of hybrid between polar and Cartesian coordinates A point is described by three coordinates: P = (r, θ, z) As usual, we view these as directions for getting to P from the origin The first coordinate tells us to travel a distance r along the X-axis; the second coordinate tells us to rotate by an angle θ around the Z-axis; the third coordinate tells us to travel a distance z parallel to the Z-axis Remark 3 As with polar coordinates, it is important that the directions be followed in this order Actually, we could actually get away with rearranging the directions as long as rotation around the Z-axis comes after movement parallel to the X-axis The translation from cylindrical coordinates to 3-dimensional Cartesian coordinates is very similar to the translation from polar coordinates to 2-dimensional Cartesian coordinates: x = r cos(θ) y = r sin(θ) z = z 3

4 215 Spherical coordinates In spherical coordinates, we write P = (R, φ, θ) Here R tells us how far up the Z-axis to go, φ tells us an angle to rotate towards the XY -plane in the XZ-plane, and θ tells us an angle to rotate around the Z-axis As with polar and cylindrical coordinates, a single point can have more than one representation in spherical coordinates Here is how to translate from spherical coordinates to cylindrical coordinates: r = R sin(φ) θ = θ We can then translate this to cartesian coordinates: 216 Exercises z = R cos(φ) x = r cos(θ) = R sin(φ) cos(θ) y = r sin(θ) = R sin(φ) sin(θ) z = R cos(φ) 1 If a point P has Cartesian coordinates ( 1, 3 1), give all ways of writing P in polar coordinates 2 Give all ways of writing the origin in polar coordinates 3 Give all ways of writing the point with Cartesian coordinates (1, 1, 1) in spherical coordinates and cylindrical coordinates 4 Give all ways of writing the origin in cylindrical and spherical coordinates 5 Write an equation describing a sphere of radius 1 in Cartesian, cylindrical, and spherical coordinates 6 Write inequalities describing a cylinder of radius 1 and height 2 in Cartesian, cylindrical, and spherical coordinates 22 Functions of multiple variables If we are interested in studying geometry, the spaces R 2 and R 3 are pretty boring We need ways to describe more interesting shapes We have already seen in the last section s exercises that shapes can be described by equalities and inequalities between formulas In this section, we will study ways of writing such formulas Recall that a function of one real variable is a rule for transforming one real number into another real number We write f : R R to mean that f is a function of one real variable We usually write f(x) for the value of f at the input x Likewise, a function of two real variables is a rule for transforming ordered pairs of real numbers into real numbers and we write f : R 2 R to indicate that f is such a function We usually write f(x, y) for the value of f at the point with Cartesian coordinates (x, y) We write f : R 3 R to mean that f is a function of 3 real variables (ie, a rule for transforming a triple of real numbers to a single real number) and we write f(x, y, z) for the value of f at the point with Cartesian coordinates (x, y, z) We won t make use of functions of more variables than 3, but these can of course be defined in the same way Example 3 The rule which sends a point with Cartesian coordinates (x, y) to the real number x 2 + y 2 is a function of two real variables If we call it f then we may say that f : R 2 R is the function given by f(x, y) = x 2 + y 2 4

5 The representation of this function in polar coordinates is f polar (r, θ) = r 2 cos(θ) 2 + r 2 sin(θ) 2 = r 2 I have written the subscript polar to emphasize that although f polar is the same function as f, it is being expressed in a different coordinate system It is also useful to study functions whose outputs are not just real numbers but are points of 2- or 3- dimensional space These functions are usually called transformations or mappings (or maps for short) since the word function is frequently used to refer only to maps whose target is R If F takes m real numbers as input and gives n real numbers as output, we say that F is a mapping from R m to R n and we write F : R m R n A mapping F : R 2 R 2 to itself is often written in terms of component functions: F (x, y) = (f(x, y), g(x, y)) This means that the x-coordinate of F (x, y) is f(x, y) and the y-coordinate of F (x, y) is g(x, y) We use similar notation for a mapping G : R n R m Example 4 Let F be the mapping given in Cartesian coordinates by F (x, y) = (y, x) This map exchanges the x- and y-coordinates of a point It is a mapping from R 2 to R 2 and we can write this succinctly: F : R 2 R 2 In polar coordinates, we could write F polar (r, θ) = (r, π 2 θ) polar We have written the subscript polar on F to indicate that we are expressing the input of F in polar coordinates and we have written the subscript polar on the output to indicate that we are also expressing the output in polar coordinates Remark 4 We could also have written F polar (r, θ) = (r sin(θ), r cos(θ)) cart This equation can be read the function F, applied to a point with polar coordinates (r, θ) is a point with Cartesian coordinates (r sin(θ), r cos(θ)) We ve expressed the input in polar coordinates and the output in Cartesian coordinates This would be a pretty confusing thing to write if we didn t specify which coordinate system we are using One of the most common ways mappings R n R n arise is from changes of coordinates For example, the translation from polar coordinates to Cartesian coordinates can be expressed by the function F Cart (r, θ) = (r cos(θ), r sin(θ)) Cart I have written the subscripts Cart to emphasize that F is the function which takes a point with Cartesian coordinates (r, θ) to a point with Cartesian coordinates (r cos(θ), r sin(θ)) A function F is called invertible if there exists a function G such that F (G(P )) = P and G(F (Q)) = Q for every Q in the domain of F and and every P in the codomain Being invertible is equivalent to being bijective, which means that 1 there are no two inputs corresponding to the same output of F, and 2 every possible output can be found from some input Remark 5 The reason there are two names for the equivalent concepts of invertibility and bijectivity is that we are usually interested in functions with special properties It is not always possible to find an inverse to a function that has the same special properties as the original function, even if the function in question is bijective 5

6 221 Level sets Now we will use functions to construct geometric objects From another perspective, we will find ways to visualize functions geometrically Suppose that F : U V is a function For any fixed element v of V, the set of u U which solve the equation f(u) = v is called a level set of F This is most frequently applied for functions whose target is R Example 5 Take the function F : R 2 R defined by F (x, y) = x 2 + y 2 The level set at height c is the set of solutions to the equation x 2 + y 2 = c That is the circle of radius c 1/2 if c 0 and empty if c < 0 Thus the level sets of F form a sequence of concentric circles around the origin in R The graph of a function If F : R n R m is a function then the graph of F is the subset of points (u, v) in R n R m = R n+m such that F (u) = v For example, if F : R R is a function of one variable, then the graph of F is drawn in the plane R 2 and consists of the points y = F (x) If F : R 2 R then the graph of F will live in R 3 In Cartesian coordinates, z = F (x, y) If F : R R 2 is given in Cartesian coordinates by F (x) = (f(x), g(x)) then the graph of F will be the set of points in R 3 with Cartesain coordinates (x, y, z) such that (y, z) = F (x), ie, such that y = f(x) z = g(x) The graph of a function is a special kind of level set If G : R n+m R m is the function G(u, v) = F (u) v then the graph of F is the level set of G at height Exercises 1 Find an inverse to the function F (x, y) = (x + y, x y) 2 Show that the function F (r, θ) = (r cos(θ), r sin(θ)) cannot have an inverse 3 Is it possible for a function from R to R 2 to be invertible? 4 Draw the graph of the function F (x, y) = x 2 + y 2 Find another function G and a value in the target of G such that the graph of F is the corresponding level set of G 5 Draw the graph of the function F (x) = (x 2, x 3 ) Express the graph of F as the level set of some function G (as in the last exercise) 23 Vectors Vectors describe how to move between points in 2- and 3-dimensional space In a sense a vector is like a set of directions for getting somewhere You could start following this set of directions starting at any point in space and you will be taken to some other point where you wind up depends on where you started Remark 6 When we learned about coordinates in the last section, we described a point in space by giving directions from the origin to that point We were really giving the vector from the origin to that point; each coordinate system we encountered gives is a different way of describing that vector 6

7 Vectors can be described in any coordinate system, just like points in space However, it is far easiest to work with vectors in Cartesian coordinates, as we will see in the exercises A vector can be expressed in 2-dimensional Cartesian coordinates with two pieces of information: how far to move parallel to the X-axis, and how far to move parallel to the Y -axis We will sometimes write v = (x, y) to mean that v is the vector with Cartesian components x (parallel to the X-axis) and y (parallel to the Y -axis) For some purposes, it is better to write the vector as a column: x v = y It should be clear how to generalize this notation to describe vectors in 3-dimensional Cartesian coordinates Suppose P and Q are points in R 2 or R 3 The vector from P to Q is written Q P If the Cartesian coordinates of P are (x 1, y 1 ) and those of Q are (x 2, y 2 ), then the Cartesian coordinates of Q P are (x 2 x 1, y 2 y 1 ) If v is a vector and P is a point then we can produce a new point by adding the vector This point is written P + v If P has Cartesian coordinates (x 1, y 1 ) and v has Cartesian coordinates (x 2, y 2 ) then 231 Vector operations P + v = (x 1 + x 2, y 1 + y 2 ) Suppose that v an w are vectors If we view each of these as a list of directions then we can produce a new list of directions by concatenating them This list is denoted v + w If v = (x 1, y 1 ) and w = (x 2, y 2 ) then This addition law is associative and commutative: (x 1, y 1 ) + (x 2, y 2 ) = (x 1 + x 2, y 1 + y 2 ) v + w = w + v u + (v + w) = (u + v) + w If λ is a real number then we can also scale the directions in v by λ: given any list of directions v, we can produce a new list of directions such that every time the original directions said to move by 1 unit in some direction, the new directions say to move by a distance λ in the same direction We will denote the vector obtained by scaling v by lambda λv If the Cartesian coordinates of v are (x 1, x 2 ) then λv = λ(x 1, x 2 ) = (λx 1, λx 2 ) This is called scalar multiplication since it scales the length of the vector; λ is therefore called a scalar (as opposed to a vector) Scalar multiplication distributes over vector addition: 232 Vectors in 3 dimensions λ(v + w) = λv + λw Vectors in 3 dimensions are very similar to vectors in 2 dimensions The only difference is that we write them with 3 coordinates 233 Standard basis vectors The simplest sorts of directions one could give are move one unit parallel to the X-axis or move one unit parallel to the Z-axis These vectors have special names: e 1 = one unit parallel to the X-axis e 2 = one unit parallel to the Y -axis e 3 = one unit parallel to the Z-axis 7

8 We will also write i for e 1, j for e 2, and k for e 3 when working in 2 or 3 dimensions Any vector in 2 dimensions can be written as a sum of multiples of i and j in just one way Any vector in 3 dimensions can be written as a sum of multiples of i, j, and k in just one way Example 6 The vector in R 3 with Cartesian coordinates (2, 3, 5) can be written as 234 Dot product, length, and projection 2i + 3j + 5k Let v denote the length of a vector v Suppose w = u + v If u and v are perpendicular then w 2 = u 2 + v 2 by the Pythagorean theorem If u and v are arbitrary then the quantity u v = 1 2 ( w 2 u 2 v 2) measures how far the triangle formed by u, v, and w is from being a right triangle, or, in a sense, how far the angle between u and v is from being a right angle This is called the dot product of u and v We have just proved the first key property of the dot product: u v = 0 if and only if u and v are perpendicular Let s calculate a formula for u v in terms of the Cartesian coordinates of u and v Suppose the Cartesian coordinates of u are (x 1, y 1 ) and those of v are (x 2, y 2 ) Then u v = 2( 1 (x1 + x 2 ) 2 + (y 1 + y 2 ) 2 x 2 1 y1 2 x 2 2 y2 2 ) = x 1 x 2 + y 1 y 2 In three dimensions a similar calculation shows that u v = x 1 x 2 + y 1 y 2 + z 1 z 2 if u has Cartesian coordinates (x 1, y 1, z 1 ) and v has Cartesian coordinates (x 2, y 2, z 2 ) One can check from the formula for the dot product that the following properties hold: u v = v u u (v + w) = u v + u w u (λv) = λ(u v) u u = u 2 One must use the Pythagorean theorem to prove the last property Suppose we want to calculate the projection of u on v That is, we want to find an expression u = λv + w where v and w are perpendicular (or orthogonal ) Using the properties above, we obtain so λ = u v = u v v 2 v v u v = λ(v v) + (w v) = λ v 2 8

9 We can also calculate using trigonometry that λ = u v cos(θ) where θ is the angle between u and v This gives us a relationship between u v and the angle between u and v: v cos(θ) = u λ = u v u v u v θ = arccos u v u v = u v cos(θ) 235 An alternate discussion of dot product, length, and projection We want to calculate the following: 1 the angle between two vectors, 2 the length of a vector, 3 the projection of one vector onto another We will see that there is a single operation, called the dot product, that can be used to calculate all three of the above If v = (x 1, y 1 ) and w = (x 2, y 2 ) in Cartesian coordinates, then v w = x 1 x 2 + y 1 y 2 If v = (x 1, y 1, z 1 ) and w = (x 2, y 2, z 2 ) in Cartesian coordinates, then v w = x 1 x 2 + y 1 y 2 + z 1 z 2 Note that the dot product of two vectors is a scalar, not a vector We will see in a moment how this is related to the properties we just described First we note some properties: If v = (x 1, y 1 ) then v w = w v (u + v) w = u w + v w v v = x y 2 1 By the Pythagorean theorem, this is the length of v, which is denoted v We therefore have v v = v 2 The same equation holds vectors in R 3 In order to calculate proj u (v) we note first that proj u (v + w) = proj u (v) + proj u (w) and proj u (λv) = λ proj u (v) Therefore we just need to figure out what proj u (i) and proj u (j) are The projection of v on u is always a multiple of u We call this multiple comp v u so that the equation proj u (v) = comp v(u) u u 9

10 holds We can calculate using trigonometry that comp v (u) = cos(θ) v where θ is the angle from u to v Likewise comp u (v) = cos( θ) u since the angle from v to u is θ Since cos( θ) = cos(θ), we have u comp v (u) = v comp u(v) Let s suppose that u = (x 1, x 2 ) Then applying what we just proved to the projection of i onto u, we get comp u (i) = 1 comp u i (u) = 1 x 1 u comp u (j) = 1 u comp j(v) = 1 u y 1 Now, suppose that the Cartesian coordinates of v are (x 2, y 2 ) so that v = x 2 i + y 2 j Then we have This gives us a formula for the projection: comp u (v) = comp u (x 2 i + y 2 j) = x 2 comp u (i) + y 2 comp u (j) = x 2 x 1 + y 2 u y 1 = u v u u proj u (v) = comp u(v) u We saw in the process of the proof that comp u v = v cos(θ) u = u v u = u v u 2 u u u where θ is the angle from u to v On the other hand, we just proved that comp u v = u v Therefore u 236 Lines and planes cos(θ) = u v u v u v = u v cos(θ) u v θ = arccos u v Let s see what the graphs and level sets of functions built from the vector operations and the dot product look like If a is any vector and b is any number, we can write down a function F (v) = a v + b In 2-dimensional Cartesian coordinates, this looks like F (x, y) = a 1 x + a 2 y + b Its graph looks like a plane in 3-dimensional space Such a function is often called linear, though it should really only be given that name if b = 0 (see Section 237 for the definition of linear); more precise names for such a function include affine linear, of first order, and of degree 1 The level sets of such a function are a family of parallel lines in 2-dimensional space We might also consider the analogous function in 3-dimensions, given in Cartesian coordinates by F (x, y, z) = a 1 x + a 2 y + a 3 z + b 10

11 Even though we cannot draw it, since it lives in a 4-dimensional space, the graph of this function is a 3-dimensional hyperplane Its level sets form a family of parallel planes in 3-dimensional space We now have two ways of representing a plane in 3-dimensional space, as well as two ways of representing a line in 2-dimensional space: as the graph of a function or as the level set of a different function If we have a plane represented in one way, how can we find a representation in the other? We ll discuss this in the next section (on cross products) Suppose now that a 1, a 2, and b are vectors in R 2 We can define a function F : R 2 R 2 by F (v) = (a 1 v, a 2 v) + b The graph of this function lives in a 4-dimensional space so we can t draw it If we could we would see a 2-dimensional plane in 4-dimensional space Such a function is also called linear Affine linear functions are the easiest functions that are not constant Calculus is essentially the study of complicated functions by linear approximation 237 Matrices and linear transformations In this section we ll do everything in R 2 first and then generalize to more dimensions Column vectors and matrices First we will introduce some notation It is very common to write vectors as columns rather than rows as we have been doing in the previous sections The reason for this is primarily historical, but it is now such a universal convention that it would be impossible to change it Thus, the components of a vector in R 2 will be written like this v = A linear transformation from R n to R m is a function with the following properties F (u + v) = F (u) + F (v) v 2 for any vectors u and v F (λv) = λf (v) for any vector v and scalar λ Since every vector can be written as a linear combination of the basis vectors e 1,, e n, a linear transformation is determined by what it does to these vectors We can therefore describe F entirely by giving a vector of n vectors in R m M = ( F (e 1 ) F (e n ) ) Notice that M is a row vector whose entries are column vectors This is called the matrix of the linear transformation F If we want to figure out what F (v) is we write down the components of v: v = v 1 v n This means that Then because F is linear, we have v = v 1 e v n e n F (v) = F (v 1 e v n e n ) = v 1 F (e 1 ) + + v n F (e n ) We ll work this out again in some special cases in a moment 11

12 Example 7 Let F be the linear transformation given by the rule + v F = 2 v 2 2v 1 3v Since e 1 = and e 0 2 =, we get F (e 1 ) = F = F (e 2 ) = F = 1 3 Thus the matrix of F is ( Applying matrices to vectors and we want to find out what F ) Suppose that F is a linear function given by the matrix a11 a A = 12 a 21 a 22 ( v 2 ) is Since ( v 2 ) = v 1 e 1 + v 2 e 2 we have F Since F is linear, this is the same thing as ( v 2 ) = F (v 1 e 1 + v 2 e 2 ) v 1 F (e 1 ) + v 2 F (e 2 ) Since F is given by the matrix A, the value of F (e 1 ) is the first column of A and F (e 2 ) is the second column of A Therefore we can rewrite the above as a11 a12 a v 1 + v a 2 = 11 v2 a + 12 a = 11 + v 2 a a 22 v 1 a 21 v 2 a 22 v 1 a 21 + v 2 a 22 Notice that the first entry of the vector on the right is the dot product of the( first ) row of A with the vector an the second entry is the dot product of the second row of A with v 2 v 2 Now that we have a rule for applying the linear transformation corresponding to a matrix to a vector, we can economize our notation: we forget about the notation and just remember how to apply a matrix to a vector The rule for applying a 2 2 matrix to a vector is summarized in the following definition Definition 1 (Applying a 2 2 matrix to a vector) If A is a 2 2 matrix and v is a vector in R 2 then the vector Av obtained by applying A to v is the one whose first component is the dot product of the first row of A with v and whose second component is the dot product of the second row of A with v More concretely, if A = a11 a 12 a 21 a 22 and v = ( v 2 ) 12

13 then ( a11 v Av = 1 + a 12 v 2 a 21 v 1 + a 22 v 2 If a 1 is used to denote the first row of A and a 2 is used for the second row then a1 v Av = a 2 v Composing linear transformations and multiplying matrices The thing that really makes matrices a useful way to think about linear functions is that there is a simple rule for figuring out the matrix of the composition of two functions from the matrices of the two functions Let s see how this works Suppose that F is a linear transformation with matrix a11 a A = 12 a 21 a 22 and G is a linear transformation with matrix This means that F G ( B = b11 b 12 b 21 b 22 ) ) a11 v = 1 + a 12 v 2 v 2 a 21 v 1 + a 22 v 2 ) b11 v = 1 + b 12 v 2 v 2 b 21 v 1 + b 22 v 2 ( To figure out the matrix of F G, we need to figure out what F G does to e 1 and e 2 The vectors F (G(e 1 )) and F (G(e 2 )) will then be the columns of the matrix of F G Now, G(e 1 ) is the first column of the matrix of G, so b11 G(e 1 ) = b 21 Let s write b 1 for the first column of B, so G(e 1 ) = b 1 We know how to apply F to this vector from the last section We multiply the matrix A with the vector b 1 and we get a1 b Ab 1 = 1 a 2 b 2 This is the first column of the matrix of F G To find the second column, we need to apply F to the second column of the matrix of G We ll write b 2 for the second column of B Then a1 b F (b 2 ) = Ab 2 = 2 a 2 b 2 This gives us the second column of the matrix of F G Let s put this together to get the matrix for the F G It is a1 b 1 a 1 b 2 a 2 b 1 a 2 b 2 Just as we did when applying a matrix to a vector, we forget about the linear transformations that we used to figure out this rule and just remember the rule It is the definition of matrix multiplication 13

14 Definition 2 (Multiplication of 2 2 matrices) Suppose that a11 a A = 12 and B = a 21 a 22 b11 b 12 b 21 b 22 are matrices Let a 1 be the first row of A and let a 2 be the second row Let b 1 be the first column of B and let b 2 be the second column of B Then the product of A and B is the matrix a1 b AB = 1 a 1 b 2 a 2 b 1 a 2 b 2 We could also write AB = ( Ab 1 Ab 2 ) By now, you may be able to guess the rule for multiplying a row vector and a matrix Using that, you can also write a1 B AB = a 2 B 238 Exercises 1 Write the rules for vector addition and scalar multiplication in polar, cylindrical, and spherical coordinates 2 Prove the law of cosines: in a triangle with sides A, B, C with lengths a, b, c, respectively, where θ is the angle between A and B c 2 = a 2 + b 2 2ab cos(θ) 3 Write the linear transformation as a matrix F (x, y) = (x + y, 2x 3y) 3 Linear approximation 4 Acknowledgements I m grateful to Jerry Park for pointing out some of my mistakes 14

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