Vectors & Coordinate Systems

Transcription

1 Vectors & Coordinate Systems Antoine Lesage Landry and Francis Dawson September 7, 2017 Contents 1 Motivations & Definition Scalar field Vector field Document contents Vector operations Scalar multiplication Addition Subtraction Linear independence and basis Linear dependence Basis Scalar product (dot product) Vector product (cross product) Scalar triple product Vector triple product Description and Representation of Coordinate Systems Representation of basis vectors Scalar product in component form Vector product in component form D coordinate systems Cartesian or rectangular Definition Differential elements Polar Definition Transformation Differential elements D coordinate systems Cartesian Definition Differential elements Cylindrical Definition Transformation Differential elements Spherical Definition c Antoine Lesage Landry and Francis Dawson, Figures by Conrad Hopp 1

2 5.3.2 Transformation Differential elements Position, position vector and general vector 34 7 Velocity and acceleration vectors in different coordinate systems D Polar coordinates D Cylindrical coordinates D Spherical coordinates Problems 40 9 Solutions 41 Appendi A Motivation for scalar triple product 43 Appendi B Addition of two position vectors in polar coordinates 44 2

3 1 Motivations & Definition Vectors are defined differently depending on the coordinate system being utilized. In any coordinate system the position vector remains the same although the coordinates used to indicate the endpoint of the vector will be different in a given coordinate system. Typically a coordinate system is chosen in order to make the calculations as simple as possible. The purpose of this document is for the students to develop the tools that enable them to perform calculations in a different coordinate system and to transfer from one coordinate system to another. You should read these notes during the first week of classes. In Tutorial 0 you will be tested on one of the assigned problems at the end of these notes. You will be assigned a grade of 0 or 1; fail or pass. This grade will not be counted but will be used to bump your final grade in the event you are close to a threshold grade, for eample, just below 40, 50, 60, 70, 80, 85, 90 or 100. You may wish to be selective in terms of what you read now but you will find this material helpful for the following two reasons: (a) the introductory material is a review of what you learned last year in linear algebra and the second calculus course (b) it provides the background material required to solve problems near the end of this term and for other courses that you may take net term and in the coming years. There are two geometrical entities which you are familiar with already. A scalar is a entity that only possesses a signed magnitude and describes an amount (a number) or an intensity (representing the magnitude of something associated with a physical entity that can be measured). For eample, we use a scalar when we epress temperature, 10 C, mass, 50kg or voltage, 12 V. Note that a scalar can be positive or negative. A vector is an entity that possesses both a magnitude and a direction at a position in space. Vectors are an important component of mathematics, physics, and engineering. In a two dimensional (2D) space the vector requires two independent directional components to describe an arbitrary direction in 2D space. In a three dimensional (3D) space, the vector requires three independent directional components to describe an arbitrary direction in 3D space. The vector can be visualized as an arrow pointing in a certain direction where the length gives the magnitude of the vector and the orientation of the arrow the direction. For eample, a vector in 3D can be used to epress a force. A person jumping up and hitting the ground applies a force of 1000 N downwards when he/she makes impact with the surface. In defining our vectors we need to first ask in how many dimensions we are making an observation. For eample a planar area has a dimension of 2 whereas the world that we live in consists of three dimensions. In representing an arbitrary vector in 2D or 3D we need to create a set of reference vectors from which any vector can be constructed. In 2D, we typically use a vector pointing in the y direction and the direction to construct an arbitrary vector in 2D. As it turns out using this arrangement is very convenient however there is no reason why vectors with orientations other than and y cannot be used. In 3D, we use a vector in the, y and z direction to construct an arbitrary vector. Later in this document we will consider other coordinate systems such as Cartesian coordinates or polar coordinates in two dimensions and cylindrical or spherical coordinates in three dimensions. Lastly, we define the notion of a scalar field or a vector field by way of an eample for each case. We etend the vector definition to multi- variable vector valued functions. In the general case, we refer to these multi-variable vector valued functions as fields. 1.1 Scalar field A scalar field is a function that for a given set of scalar inputs representing for eample a position in space, returns a scalar output. The temperature distribution in a room, e.g. T (, y) in (1) and represented in Figure 1, is a scalar field where we enter the position in 2D, e.g. (1, 1), and it returns a scalar, the temperature T (1, 1) = T (, y) = cos 2 + cos y 3 (1) 3

4 2 1.8 T(,y) y -2-2 Figure 1: T (, y) scalar field 1.2 Vector field A vector field is a function that returns a vector with respect to a position denoted by scalar coordinate values. For eample, the electric field E in 3D due to a point charge Q given in (2) is a vector field. E(, y, z) = Q ( î + y ĵ + z 4πɛ 0 ( 2 + y 2 + z 2 ) ˆk ) 3/2 For every position, a different vector with its own magnitude and direction is generated. A pictorial representation of the vector field (2) is shown in Figure 2. This vector field returns, when evaluated at position (1, 1, 1), E(1, 1, 1) = Q( î+ ĵ+ ˆk) 4πɛ 03, a vector, where ɛ 3/2 0 is a physical constant. 1.3 Document contents We start by reviewing some important definitions and aioms associated with vectors, scalar products, vector products and triple products in Section 2. Once the vectors are introduced, we present coordinates systems and revisit the scalar and vector product. Section 3 discusses the two main coordinate systems in 2D: Cartesian and polar coordinates. We then etend the discussion to 3D coordinates systems and discuss Cartesian, cylindrical and spherical coordinates. Section 6 discusses the difference between a position, a position vector, and a general vector. Section 7 presents the velocity and the acceleration vectors in polar, cylindrical and spherical coordinate system. Section 8 gives problems related to the topics covered in this document. Their solutions are given in Section 9. Finally, complementary discussions on the scalar product and on the sum of two vectors described in non-cartesian coordinate systems is presented in Appendices A and B respectively. These notes are based on the introductory chapters of [1] and [2]. 2 Vector operations Let v be for eample a vector in 2D. v is visualized as an arrow, and the symbol v, a scalar, defines its magnitude and represents its length. Alternatively, we may wish to know only the direction of the vector (2) 4

5 z y Figure 2: E(, y, z) vector field independent of its length in which case we normalize the vector s magnitude to 1. The normalized vector is referred to as a unit vector ˆv and is defined as follows: ˆv = v v. Given a unit vector we can always recover the original vector by multiplying it by the magnitude of the original vector that is v = v ˆv. (3) We now define and illustrate several vector operations. 2.1 Scalar multiplication By scalar multiplication, we refer to the case where we are interested in r = αv with α R, a scalar. The scalar multiplication is a scaling operation: it modifies the magnitude of the vector. When picturing vectors as arrows, the scalar multiplication has the effect of lengthening (α > 1) or shortening (0 < α < 1) the arrow without changing the direction in which it points. Lastly, the scalar product of α with a vector for α < 0 reverses the direction of the vector, i.e. reverses the orientation of the arrow. Scalar multiplication is summarized in Figure Addition Let u be a second vector and consider a third vector r as the addition of the vectors u and v that is r = u+v. The addition of two vectors can be represented graphically by placing the tail of a vector to the head of the second one. Then, the resulting r vector is given by the vector between the tail of the initial vector and the head of the second one. Figure 4b summarizes this process. Another technique for computing the sum of vectors is the parallelogram approach. This is done by first placing both vectors tail to tail. Then, we complete the parallelogram by drawing from the head of each vector a translated version of the two initial vectors. The heads of these two vectors converge on a common 5

6 v v v v Figure 3: Scalar multiplication of vectors u v r v u (a) Vector v and u (b) Tail to head approach u v r v u u v r (c) Parallelogram approach (d) Commutativity Figure 4: Addition of two vectors point. The resulting vector r is given by the diagonal of the parallelogram passing through both initial vector tails and the translated vector heads. Figure 4c summarizes this alternative technique. Properties. The Addition of vectors is commutative: u+v = v+u and associative: (u+v)+w = u+(v+w) where w is also a vector. 2.3 Subtraction The subtraction of two vectors differs from scalar subtraction. Consider r = v u. To evaluate r, we rewrite r = v + ( u). In other words, the subtraction of two vectors is the sum of a vector and 1 time the second 6

7 one. An eample for r = v u is given in Figure 5. v r ( 1) u Figure 5: Subtraction of two vectors 2.4 Linear independence and basis Knowing the operations we can perform using vectors we can now define the concept of linear independence Linear dependence Consider the set of vectors v 1, v 2,..., v n. Then, this set of vectors is linearly independent if and only if a 1 v 1 + a 2 v a n v n = 0 (4) under the condition that a i = 0 for i = 1, 2,..., n where n represents the dimension of the space. So for eample in 2D, n = 2 and in 3D, n = 3. Alternatively, a set of n dimensional vectors is linearly independent if none of these vectors can not be rewritten as a linear combination of the remaining n 1 vectors. Figure 6 gives a few eamples of linearly dependent and independent vectors in 2D and 3D. We will consider the eample in Figure 6b for illustration purposes and show that the vectors are linearly dependent. First, we observe that v 3 = v 1 + v 2. Hence, there eists a vector in the set {v 1, v 2, v 3 } that can be reepressed as a linear combination of the remaining two vectors. We can conclude that these vectors are not linearly independent. Another way to show their linear dependence is using (4) and a non-zero a i coefficient. By setting, a 1 = 1, a 2 = 1 and a 3 = 1, we have, v 1 v 2 + v 3 = 0. (5) Figure 7 demonstrates this last result using a graphical approach based on the results demonstrated in Sections 2.2 and Basis Lastly, we define a basis as a set of linearly independent vectors that can span the whole vector space in which the vectors eist. In others words, a set of vectors forms a basis if using linear combination, we can span any vectors in the space under consideration (e.g. 2D and 3D space). The choice of basis vectors is not unique and the number of basis vectors required to span a space is equal to the dimension of that space. Later we will show that an orthogonal set of basis vectors is useful from a practical perspective. For eample, in the 2D case, we need two vectors, v 1 and v 2, to establish a basis and thus any vector v can be epressed as a linear combination of basis vectors; e.g. v = av 1 + bv 2. In 3D, we need a set of three vectors to form a basis and thus v = av 1 + bv 2 + cv 3. Note, a,b and c are constants and are real numbers. 2.5 Scalar product (dot product) The scalar product or dot product between two vectors is denoted by v u. Let s first start by considering v û, where û is the unit vector in the direction of u. Then, the scalar product v û represents the projection 7

8 v 1 v 1 v 3 v 2 v 2 (a) Linearly independent vectors in 2D (b) Linearly dependent vectors in 2D v 1 v 3 v 2 v 2 v 3 v 1 (c) Linearly independent vectors in 3D (d) Linearly dependent vectors in 3D Figure 6: Linearly dependent and independent vectors v 3 v 1 v 2 Figure 7: Graphic computation of (5) of the v vector on û. We can interpret projection v û as computing how much of v is directed in the û hat direction. To illustrate a projection, picture a vector v centered at the origin presented in Figure 8. We are then interested in the projection of this vector onto a vector û along the ais. The projection of v represents its -component given by v cos θ vu. Observe that output of the scalar product and a projection is a scalar. Now, let s state the formal definition of the scalar product where we put no restriction on the form of the vector u. v u = v u cos θ vu (6) where θ vu is the smallest angle between v and u (cf. Figure 8). Hence, for the general case, the scalar product of any two vectors, v u, represents a scaled projection of one vector onto the second vector. The scaling factor is given by the magnitude of the second vector. In the definition 6, we retrieve the projection part with v cos θ vu and the scaling factor is the magnitude of u. Note that the other way around is also true: 8

9 v û uv v u Figure 8: Projection of v onto u this can represent the projection of u along v, that is u cos θ vu, scaled by the magnitude of v. The scalar product v u is equivalent to the scalar product u v therefore the scalar product is commutative. The angle θ vu is defined as the smallest angle between the two vectors and hence can only be in the range [0, π]. This implies that the result of a scalar product can be either positive or negative depending on the angle. Consider the following three special cases: a) if θ vu = π 2, the vectors are perpendicular and using our previous analogy, there is no u-component in v; b) if θ vu = 0, the scalar product reduces to v u = v u, the product of the magnitude of the two vectors. c) if u = v, we observe the following: v v = v 2, the square of the magnitude of v. This definition of the magnitude in terms of the scalar product holds for all coordinate systems and hence provides a general approach for computing the magnitude of a vector. Properties. The scalar product is commutative: v u = u v. The scalar product is also distributive : w (v + u) = w v + w u. 2.6 Vector product (cross product) The vector product v u is a vector operation that is only valid in 3D. It takes two vectors as inputs and generates a third vector. This third vector is, by definition, perpendicular to both input vectors and is normal to the plane described by the two vectors. The direction of the resulting vector is given by the right-hand rule that goes as follows: place the edge of your right hand along the first vector, v and close your fingers in the direction of the smallest angle between the two vectors. Then, the thumb will point in the direction of the resultant vector. We denote ˆn as the unit vector normal to the plane formed by the two vectors in the vector product. Figure 9 shows graphically the orientation of ˆn for the vector product v u and u v. Notice that the orientation of ˆn is reversed when the order of the vectors u and v are reversed. The formal definition of the vector product is: v u = v u sin θ vuˆn (7) In this case, we note that if v and u are parallel, then θ vu = 0 and the cross product is zero. The magnitude of v u also has a geometric interpretation. It represents the area of a parallelogram formed by the two vectors placed tail to tail as represented in Figure 10. To see this, we first note that v u = v u sin θ vu ˆn = v u sin θ vu, since the magnitude of ˆn is one. We analyze v and u sin θ vu separately. First, the magnitude of v represents the base of the parallelogram. Then, trace a right-angled triangle ABC as shown in Figure 10. Using the definition of sin, u sin θ vu represents the length 9

10 u vu v ˆn for v u (a) v u and ˆn ˆn u vu into the page v for u v (b) v u and ˆn Figure 9: Vector product A u uv B C u v v Figure 10: Projection of v onto u of the opposite side in the triangle, and is the the height of the parallelogram. The area of a parallelogram is given by the product of its height and base and hence we get our result. Properties. The cross product is anti-commutative: v u = u v. We can derive this relation using the right-hand rule. The angle θ vu stays the same, but the direction of the thumb and hence of ˆn is reversed and we find the anti-commutative relation. The vector product is distributive over addition: w (v + u) = w v+w u. It is not associative : w (v u) (w v) u. This last statement can be shown graphically by only considering the direction of each cross product. For eample consider two nonorthogonal vectors u and v lying in a plane and w normal to this plane. Then, the left-hand side of the statement is 0 whereas the right-hand side is not zero. 2.7 Scalar triple product The triple product is a vector operation defined over three input vectors. There are two types of triple product: a scalar triple product and a vector triple product. The vector triple product will be covered in the net section. The scalar triple product, as its name suggests, generates a scalar output. The scalar triple product takes the form w (v u). The scalar triple product, like the vector product, can be geometrically interpreted. The absolute value of the scalar triple product represents the volume of a parallelepiped (3D parallelogram) formed by the three vectors placed tail to tail to tail. This parallelepiped is presented in Figure 11. To show this, we first apply the vector and scalar product using (6) and (7) respectively: v u = ˆn v u sin θ uv, (8) w (v u) = w v u cos φ w,n. (9) Here ˆn is a unit vector normal to the u,v-plane, θ uv is the angle between u and v and φ w,n denotes the angle between w and the normal ˆn. The norm of (8) is, v u = v u sin θ uv 10

11 u uv wn v Figure 11: Parallelepiped formed by w, u and v ˆn n since ˆn is a unit vector. Substituting this result in (9) results in, w (v u) = w cos φ w,n v u sin θ uv We now have two distinct terms: w cos φ w,n and v u sin θ uv. In the last section, we showed that v u sin θ uv is equal to the area formed by the parallelogram. In this case, this result gives us the area of the base of the parallelepiped. The scalar product term w cos φ w,n is equivalent to the projection of w onto the direction of ˆn, i.e. the magnitude of the ˆn component of w, which represents the base of the parallelepiped. Hence, the product of the two terms gives the volume of the parallelepiped formed by w, v and u. Properties. The scalar triple product is invariant under cyclic permutation, that is, w (v u) = v (u w) = u (w v). Since the vector product is anti-commutative, the scalar triple product is also anti-commutative with respect to the two vectors comprising the vector product: w (v u) = w (u w) = v (w u) = u (v w). The motivation for the use of the scalar triple product aside from its geometric interpretation is described in Appendi A. 2.8 Vector triple product The second type of triple product is the vector triple product. It won t be used in this course so we only define it briefly and omit the proof of the net identity. The vector triple product outputs a vector and is defined as w (v u). The vector triple product can also be re-epressed according to the following vector identity, w (v u) = v(w u) u(w v). 3 Description and Representation of Coordinate Systems Thus far, we have looked at vectors from a general perspective by sketching them as directed arrows in space. We can now introduce the tool to represent them according to basis vector components which are defined using a coordinate system. A coordinate system of a n dimensional space is a system at which a point in space is defined by the intersection point of n objects having an (n 1) dimensional form. For eample, a line has a 1 dimensional form whereas a surface has a 2 dimensional form. In 2D, this means the intersection of 2 lines whereas in 3D this represents the intersection of three surfaces. The (n 1)-dimensional objects also reveal the coordinate direction or vectors that form the basis of a system which we refer as the base vectors. As mentioned previously, base vectors are linearly independent vectors spanning the whole space of interest 11

12 and hence can span any vectors included in the n dimensional space. There are no restrictions about their relative orientation or norm at this point. It is however convenient to choose the base vectors as unit vectors that are all orthogonal with respect to each other in which case we have an orthonormal basis. Recall that orthogonality implies that the angle between any two base vectors is 90 degrees and hence their scalar dot product is zero. 3.1 Representation of basis vectors In a coordinate system, the base vectors are defined as the directions perpendicular to the (n 1)-dimension objects. With a coordinate system, a position is specified by an n-tuple made of the parameters of the (n 1)- dimensional objects specific to the system. Then, arbitrary vectors are rewritten as a linear combination of the base vectors of the coordinate system. Finally, the base vectors point in the increasing direction of an ais and to fi the proper direction, we need to impose the constraint of a right-handed system. A system is called right-handed if it satisfies a property that can be verified by the right hand rule. To check this condition, we first order the base vectors and then we place our inde finger along the first direction of the base vector and our middle finger along the direction of the second base vector. Then, if the system is right-handed, the thumb will point in the direction described by the third base vector. The order of the unit base vectors in a Cartesian system is given by convention in terms of an ascending order, i.e. ˆ-ŷ-ẑ. A more formal definition will be given after orthogonal and orthonormal coordinate systems are introduced. In summary, the base vectors we consider here are unit vectors perpendicular to their respective (n 1)- dimensional objects. Since these objects are perpendicular to each other at a given point, the unit vectors are also orthogonal to each other. Hence, a 2D orthonormal coordinate system with base vectors ˆn 1 and ˆn 2 will satisfy the following conditions, ˆn 1 ˆn 1 = 1 ˆn 2 ˆn 1 = 0 ˆn 1 ˆn 2 = 0 ˆn 2 ˆn 2 = 1, and a 3D orthonormal coordinate system with base vectors ˆn 1, ˆn 2 and ˆn 3 will satisfy the following conditions: ˆn 1 ˆn 1 = 1 ˆn 2 ˆn 1 = 0 ˆn 3 ˆn 1 = 0 ˆn 1 ˆn 2 = 0 ˆn 2 ˆn 2 = 1 ˆn 3 ˆn 2 = 0 ˆn 1 ˆn 3 = 0 ˆn 2 ˆn 3 = 0 ˆn 3 ˆn 3 = 1, where we used ˆn i to denote the fact that the base vectors are normalized. However, note that there is nothing which keeps us from using a non-orthogonal coordinate system. For eample, a non-orthogonal coordinate system can be used in crystallography when working with a triclinic or monoclinic lattice. The monoclinic system is sketched in Figure 12. In this case, the angle between the three base vectors associated with the crystallographic aes are α = π/2, γ = π/2 and β π/2 making this system non-orthogonal. v 3 c 2 v 1 v 2 a bac Figure 12: Monoclinic system b 12

13 For an orthonormal 3D coordinate system, the right-handed system condition can be defined formally using the cross product of its base vectors. The base vectors with the ordered sequence ˆn 1, ˆn 2 and ˆn 3 of an orthonormal right-handed system must satisfy the following condition: ˆn 1 ˆn 2 = ˆn 3 ˆn 2 ˆn 3 = ˆn 1 (10) ˆn 3 ˆn 1 = ˆn 2 which is equivalent to the right-hand rule. In the two net sections, we will study the most common orthonormal coordinate systems. Our goal is to epress any vector in the form of a linear combination of the base vectors. In the nd case where n represents the number of dimensions, this means that we will epress a vector v as v = v 1ˆn 1 + v 2ˆn v nˆn n, where ˆn i i = 1, 2,..., n are the base vectors for an orthonormal basis defined by the coordinate system and a i i = 1, 2,..., n are the weights of each base vector. This representation of a vector is referred to the component form. In the 3D case we have, v = v 1ˆn 1 + v 2ˆn 2 + v 3ˆn 3. (11) Consider a Cartesian coordinate system in which case we can denote the orthonormal basis vectors as, ˆn 1 = 1 0, ˆn 2 = , ˆn 3 = (11) is equivalent to, v 1 v = ˆn 1 ˆn 2 ˆn 3 v 2 v 3 = v 1 v v 3 Alternatively, we can represent the v vector in a single column form as, v 1 v = v 2, v 3 where the base vectors are taken to be implicitly ˆn 1, ˆn 2 and ˆn 3 and the representation of the unit base unit vectors depends on the coordinate system. The goal of the coming section is thus to define the ˆn i needed to represent a vector in a coordinate system other than Cartesian Scalar product in component form We now revisit the scalar product using the component form in an orthonormal 3D coordinate system. First, we relate the scalar product formal definition (6) to the component form of the scalar product in the case of an orthonormal system using a geometric interpretation. We demonstrate the relation only in 2D for Cartesian coordinates (cf. Section 4.1) but it can also be etended to higher dimensions and other coordinate systems. Consider the vectors v and u represented in Figure 13. u v = u v + u y v y (12) 13

14 y u v 1 2 Figure 13: Relation between definitions of the scalar product The dot product u v can be re-epressed as, u v = u v cos θ uv = u v cos (θ 1 θ 2 ) = u v (cos θ 1 cos θ 2 + sin θ 1 sin θ 2 ) (13) Using Figure 13 and denoting u, u y and v and v y as the -component and y-component of u and v respectively we can observe the following relations: cos θ 1 = u u, cos θ 2 = v v, sin θ 1 = u y u, sin θ 2 = v y v. We can therefore rewrite (13) as, ( u u v = u v u = u v + u y v y, v v y v + u y u v and we retrieve the component form of the scalar product. In 3D, the angle θ is between vector u and v on the plane defined by the vectors u and v. Performing the same calculations, we get, ), u v = u v cos θ = u v + u y v y + u z v z Finally, we can etend this to spaces with more than three dimensions. Then, following our introduction of the component form of the scalar product with the geometric interpretation, we can consider the general case of the dot product of two arbitrary 3D vectors epressed in terms of an orthonormal basis where, then, v = v 1ˆn 1 + v 2ˆn 2 + v 3ˆn 3, u = u 1ˆn 1 + u 2ˆn 2 + u 3ˆn 3, v u = (v 1ˆn 1 + v 2ˆn 2 + v 3ˆn 3 ) (u 1ˆn 1 + u 2ˆn 2 + u 3ˆn 3 ), = v 1 u 1ˆn 1 ˆn 1 + v 1 u 2ˆn 1 ˆn 2 + v 1 u 3ˆn 1 ˆn 3 + v 2 u 1ˆn 2 ˆn 1 + v 2 u 2ˆn 2 ˆn 2 + v 2 u 3ˆn 2 ˆn 3 + v 3 u 1ˆn 3 ˆn 1 + v 3 u 2ˆn 3 ˆn 2 + v 3 u 3ˆn 3 ˆn 3 (14) 14

15 We recall the definition of the dot product, notably that if two vectors are perpendicular, then θ = π/2 and the dot product is zero and if they are parallel, θ = 0 and the dot product is equal to the product of the norms. In this case, the norms are all 1 since ˆn i are unit vectors. Hence, (14) reduces to v u = v 1 u 1 + v 2 u 2 + v 3 u 3. (15) This last form is valid for all orthogonal coordinate systems and can be generalized to a larger number of dimensions in space. This also provides one justification of why orthonormal coordinate systems are easier to work with: there are no cross terms in the dot product epansion. 3.3 Vector product in component form We can also give an alternate definition of the vector product in component form in 3D. We consider v and u as previously defined and compute the cross product between the two vectors, v u = (v 1ˆn 1 + v 2ˆn 2 + v 3ˆn 3 ) (u 1ˆn 1 + u 2ˆn 2 + u 3ˆn 3 ), = v 1 u 1ˆn 1 ˆn 1 + v 1 u 2ˆn 1 ˆn 2 + v 1 u 3ˆn 1 ˆn 3 + v 2 u 1ˆn 2 ˆn 1 + v 2 u 2ˆn 2 ˆn 2 + v 2 u 3ˆn 2 ˆn 3 + v 3 u 1ˆn 3 ˆn 1 + v 3 u 2ˆn 3 ˆn 2 + v 3 u 3ˆn 3 ˆn 3 (16) In an orthonormal right-handed coordinate set, we can use three properties to compute (16): (i) the cross product of a vector with itself is zero, (ii) the anti-commutative property of the cross product, (iii) the right-handed condition (10). The vector product can be computed in the following way, Rearranging (17), we have, Finally, (18) is equivalent to, v u = v 1 u 2ˆn 3 v 1 u 3ˆn 2 v 2 u 1ˆn 3 + v 2 u 3ˆn 1 + v 3 u 1ˆn 2 v 3 u 2ˆn 1. (17) v u = (v 2 u 3 v 3 u 2 ) ˆn 1 (v 1 u 3 v 3 u 1 ) ˆn 2 + (v 1 u 2 v 2 u 1 ) ˆn 3. (18) ˆn 1 ˆn 2 ˆn 3 u v = u 1 u 2 u 3 v 1 v 2 v 3 = M, where M represents the determinant of M. This concludes our introduction of vectors and coordinate system. Let s now eplore the common orthogonal coordinate systems. 4 2D coordinate systems We begin our coordinate system survey by considering the two dimensional case. In 2D, we will cover Cartesian and polar coordinate systems. For all coordinate systems, we will discuss how positions are represented, what the base vectors and vector representation are and how to convert from one coordinate system to another. Finally, we will discuss differential length, surface and volume as we will need these tools later in the course. 4.1 Cartesian or rectangular Definition The Cartesian coordinate system (sometimes referred as rectangular in 2D) is the most common system. It is also the easiest to work with, but may on occasion be awkward to use because the symmetry of the problem may suggest the need for an alternative coordinate system to simplify calculations. For eample, consider any point at a certain distance from the origin having the same property (e.g. magnitude of potential is constant). This means that all points lying on a circle of a given radius are equivalent when computations are made and thus we have a symmetry commonly referred to as a polar symmetry. 15

16 As described earlier, a position in 2D is described by the intersection of two lines in a Cartesian space. The first line is the line = 0 and the second line is the line y = y 0. We first observe that the two lines are perpendicular and hence the 2D Cartesian system is orthogonal. Both 0 ], + [ and y 0 ], + [ are constant and taken together represent the position of a point P,where and y are each aligned with one of the two base vectors. This is represented in Figure 14. With these two lines for varying 0 and y 0, we can represent any point in a 2D space. y ŷ y y o ˆ o Figure 14: Cartesian coordinate system As introduced in the previous section, the unit direction and base vectors of a 2D system are given by a unit vector perpendicular to the (n 1) dimensional object. In a 2D Cartesian, this means that one base vector is perpendicular to the line = 0 and the second base vector is perpendicular to the line y = y 0. However, that leaves us with two possible orientation for each base vector. In order to fi the orientation, we recall the definition of a right-handed system. By convention, the ordered base vectors are {ˆ, ŷ}. We then align the hand with the ˆ direction and then the fingers point along the direction of ŷ so it looks like a counterclockwise rotation with reference to the ais in the y plane. The two base vectors denoted as ˆ and ŷ are represented in Figure 14 and indicate the increasing dimension of each ais. Finally, we can re-write any vector v R 2 as v = v ˆ + v y ŷ where v, v y are two constants. These constants represent the amount of ˆ-vector and ŷ-vector needed to construct v. They hence represent the projection of v onto ˆ and ŷ respectively. Figure 15 gives an eample of this representation. y ŷ r ˆ 5y ˆ 3ˆ Figure 15: Vector decomposition in Cartesian 16

17 4.1.2 Differential elements We now investigate the representation of the differential geometric elements in a Cartesian coordinate system. The differential geometric elements; length and surface area, can be found by looking at the change due to an infinitesimal variation in every direction associated with the base vectors. We start by computing the differential length dl i, i =, y. From that, we can obtain two things, (i) the vector representing differential length for an arbitrary orientation and (ii) the differential surface area. y y d y ds ddy d ŷ dy ˆ d d Figure 16: Differential elements in Cartesian For the Cartesian case, Figure 16 gives a representation of the infinitesimal variation along the ˆ and ŷ direction. From the figure, we deduce that dl = d and dl y = dy. Hence, the vector representing the differential length is dl = dl ˆ + dl y ŷ Net, the differential surface area ds is given by the area of the rectangle created by the variations along the direction of the basis vectors. Hence, we have ds = dl dl y or, ds = ddy 4.2 Polar Definition Like all 2D coordinate systems, the polar coordinate system is described by two lines. In a polar coordinate system, the first line is the circle defined by r = r 0. The variable r is taken to be greater than or equal to zero. This circle represents all the points located on a line which is at a distance r 0 from the origin, in other words all (, y) points that satisfy the equation 2 + y 2 = r 0. We can also see this circular line as the union of two half circular lines: f() = r and f() = r for [ r 0, r 0 ]. The second line is the straight line going through the origin and making an angle θ = θ 0 with respect to the ais measured in a counterclockwise direction. Figure 17 shows the two lines that define the polar coordinate system. Hence, any point in 2D can be epressed as a pair (r, θ) where r [0, + [ and θ [0, 2π] as shown in Figure 17. We observe that for any θ 0 and r 0, the line θ = θ 0 and the circle r = r 0 are perpendicular, thus making this system an orthogonal system. Then, the base vectors of this system are defined by unit vectors oriented perpendicularly to the two lines. Once again, both unit vectors could have two orientations: inward and outward for ˆr and clockwise and counterclockwise for ˆθ. In this case, we first order the two base vectors as 17

18 {ˆr, ˆθ} and set ˆr to point outward so that the r component increases when going further away from the origin. This has a geometric meaning since r represents the distance of a point from the origin which increases as we go further away from the origin. Then placing our hand along the direction of ˆr and closing it, our fingers point in the counterclockwise direction which gives the direction of ˆθ. Setting ˆr and ˆθ this way makes the 2D coordinate system right-handed. Finally, the base vectors ˆr and ˆθ for the polar coordinate system are represented in Figure 17. In conclusion, in a right-handed polar coordinate system, the base vector or unit vector ˆr is radial, in other words, from the origin it points outwards and the base vector ˆθ is oriented counterclockwise and is always oriented tangential to a circle that intersects the point (r 0, θ 0 ). We can write any vector v R 2 in a polar coordinate system in the form of v = v rˆr + v θ ˆθ, where the coefficients v r, v θ are given by the projection along the ˆr and ˆθ direction. y ˆθ r ˆθ ˆr (, y) ( r, ) ˆr Figure 17: Polar coordinate system y B Pr (, ) r r sin( ) A r cos( ) c Figure 18: Relation between position in Cartesian and polar coordinates Transformation We now want to convert a position P (r, θ) and a vector v = v rˆr + v θ ˆθ from polar coordinates to Cartesian coordinates. The former is straightforward and only uses trigonometry. The latter, however, needs a bit more work and we detail our approach later on and re-use it in the sections to follow. Given a point P (r, θ) as shown in Figure 18, we can trace the right-angled triangle ABC and use trigonometry to find the base and height y of this vector. Recalling the definition of sine and cosine, we obtain the 18

19 following relation to convert from polar coordinates to Cartesian coordinates: = r cos θ y = r sin θ (19) Or inversely, from polar coordinates to Cartesian coordinates: r = 2 + y 2 θ = arctan y (20) Note that arctan can have two possible values for the same ratio y/. Using (20) and (19) we convert from one position in one coordinate system to an equivalent position in the other coordinate system. We now want to convert a vector representation in one coordinate system to a new vector in a different coordinate system. We need to define a mapping between a vector epressed in one coordinate system to a vector in the other coordinate system. For vectors, this mapping is done using a transformation matri. Hence, we need to determine the proper transformation matri which we will refer to as M. The objective will be to determine the elements of the transformation matri. Denote v as a 2D vector and denote v cart = v ˆ + v y ŷ as the vector epressed in Cartesian coordinates and v pol = v rˆr + v θ ˆθ as the vector epressed in polar coordinates. Note that v = vcart = v pol even though they are epressed using different basis vectors associated with the different coordinate systems. Let s first start by converting v pol into v cart. The form of the desired equation is Or equivalently, [ v v cart = Mv pol (21) ] [ ] [ ] M11 M = 12 vr v y M 22 v θ M 21 Now, recall how we obtained v in the previous section. It is the projection of v cart onto the base vector ˆ. Equivalently since v cart = v pol, v = v pol ˆ Similarly for v y, we have v y = v pol ŷ. In summary,, we have, v = v pol ˆ v y = v pol ŷ We replace v pol in the two previous equations above by their equivalent equations in polar coordinates. The set of equations becomes: ( ) v = v rˆr + v θ ˆθ ˆ ( ) v y = v rˆr + v θ ˆθ ŷ or, v = ˆr ˆv r + ˆθ ˆv θ v y = ˆr ŷv r + ˆθ ŷv θ We see that we can rewrite this equation in the form of (22) and we obtain [ v ] [ˆr ˆ = v y ˆr ŷ ] [ ] ˆθ ˆ vr ˆθ ŷ v θ where each matri element is the projection of a unit vector of one coordinate system onto the other coordinate system. In a 2D system there are 4 possible combinations representing the 2 by 2 matri. We will see later (22) 19

20 ˆθ ˆr cos( ) 2 2 cos( ) ŷ sin( ) 2 2 sin( ) ˆ Figure 19: Relation between Cartesian and polar vectors that this can be generalized to 3D to obtain a similar form of transformation matri from cylindrical to Cartesian or spherical to Cartesian but in the form of a 3 by 3 matri. The net step is to evaluate the dot product of each combination of unit vector. To do so, we refer to Figure 19. We find, ˆr ˆ = cos θ ˆr ŷ = sin θ ˆθ ˆ = cos π 2 θ = sin θ ˆθ ŷ = sin π 2 θ = cos θ Finally, we have, [ v v y ] [ ] [ ] cos θ sin θ vr = sin θ cos θ Therefore, the transformation matri from polar coordinates to Cartesian coordinates is [ ] cos θ sin θ M = sin θ cos θ v θ (23) Finally, from (23), we deduce that v = v r cos θ v θ sin θ v y = v r sin θ + v θ cos θ There eists an alternative to this approach where we focus our effort on re-epressing the base vector of one system in terms of the base vector for the new system. For this eample, we continue working on converting a polar vector into a Cartesian vector. We will discuss how this is almost equivalent to converting a Cartesian vector into a polar vector shortly after this demonstration. Our starting point is the vector in polar form which we re-state here v = v pol = v rˆr + v θ ˆθ (24) Our approach consists of finding an epression for both ˆr and ˆθ in terms of ˆ and ŷ. We first define the position vector describing the vector going from the origin to any point (, y): r = ˆ + yŷ, 20

21 and we use the polar relations (20) to re-epress the coefficient of the equation in terms of (r, θ). To relate the polar base vector, we want to keep this equation in terms of the Cartesian base vectors. We now have, r = r cos θˆ + r sin θŷ The base vectors for the polar coordinate system can be computed by taking the derivatives of the position vector with respect to a new set of variables. See Chapter 12, section 12.6 of your tetbook [3] for more details. The base vectors represent the direction of an infinitesimal change in one variable as shown in Figure 21. Finally, since we work with an orthonormal system, we normalize each direction to get a pair of unit vectors. r(n) r(n ) r(n) nˆ lim rn ( ) n 0 nn r(n n) Figure 20: Base vectors definition in terms of limits Formally, the definition of a base vector in an orthonormal system is dr dn i ˆn i = dr (25) dn i where r is the position vector epressed in terms of variables n i, where i = 1,..., n represent the indices for the individual basis vectors of the final coordinate system and n represents the number of dimensions. For a polar coordinate system, we obtain the following epressions for the base vectors ˆr and ˆθ: ˆr = ˆθ = dr dr dr dr dr dθ dr dθ,. (26) (27) These epressions are represented in Figure 21 where the derivatives are epressed in terms of limits. We first evaluate the derivative with respect to r using (26): with a norm given by, The first orthonormal base vector is therefore, dr = cos θˆ + sin θŷ dr dr dr cos = 2 θ + sin 2 θ = 1 ˆr = cos θˆ + sin θŷ = r ˆ + r y ŷ. 21

22 ˆr r( ) r r() r r( ) r( ) r( ) r( ) (a) ˆr (b) θ = r(θ + θ) r(θ) r( + ) ˆ r( ) (c) ˆθ Figure 21: Definition of the base vectors ˆθ in terms of limits We proceed the same way for ˆθ using (27) and obtain, and with a norm given by dr = r sin θˆ + r cos θŷ dθ dr dθ r = 2 sin 2 θ + r 2 cos 2 θ = r 2 = r. The second orthonormal base vector is therefore, ˆθ = sin θˆ + cos θŷ = θ ˆ + θ y ŷ. Now, we substitute these two relations in (24) and we get, v = v r (cos θˆ + sin θŷ) + v θ ( sin θˆ + cos θŷ) = (v r cos θ v θ sin θ) ˆ + (v r sin θ + v θ cos θ) ŷ 22

23 We re-write v in Cartesian coordinates as, v cart = v ˆ + v y ŷ [ ] v =, and finally obtain the the transformation matri as given in (23), [ ] [ ] [ ] v cos θ sin θ vr = v y sin θ cos θ v θ By inspection, we observe that and hence, [ v v y ] v y = [ˆr [ ] v r ˆθ] v θ [ˆr M = ˆr y where ˆr, ˆr y and ˆθ, ˆθ y are the,y component of ˆr and ˆθ respectively and form M, the transformation matri. We can generalize this approach for any orthonormal coordinate system in any dimension. The general transformation matri has the following form,... M = ˆn 1 ˆn 2... ˆn n... In summary, to convert a vector from one set of coordinates to another, we need to find the transformation matri M. Once this is done, we use matri multiplication to obtain the vector in the desired form. This is, however, only one of many transformations that we could think of. One might be required to convert a Cartesian vector to a polar vector. The procedure is to recall (21) and solve for v cart : ] ˆθ ˆθy v cart = Mv pol M 1 v cart = M 1 Mv pol v pol = M 1 v cart (28) At this point it is useful to note the following property for M. The transformation matri between two orthonormal coordinate systems is orthogonal. An orthogonal matri means that each of its columns represent the orthonormal base vectors. From previous courses you may recall that the following condition is satisfied for an orthonormal matri: MM T = I. This epression implies M T = M 1. Hence we can retrieve the polar vector using the transpose of the matri M that we developed before, v pol = M T v cart Finally, let us obtain one more useful characterization of the orthogonality of M by looking at the determinant of M. We recall that for any matri A R p p, det A = det A T. We note that det I = det ( M T M ) = det ( M T) det(m) = (det M) 2 where the determinant of the identity matri is 1. M being orthogonal also implies that its determinant is ±1. Then, computing the determinant of the matri allows one to confirm that there was no computation error. However, if the determinant is ±1, then it does not imply that we have the correct matri. Let s conclude our discussion on the transformation matri by proving that the determinant of M is 1. We have, det M = cos θ sin θ sin θ cos θ = cos 2 θ + sin 2 θ = 1 23

24 4.2.3 Differential elements The net important topic to discuss when introducing a new coordinate set is how to define the differential length dl, a vector and the surface area ds, a scalar, both in 2D. This is obtained by looking at an infinitesimal change along the directions of each base vector. Formally, dl = dl rˆr + dl θ ˆθ, ds = dl r dl θ. The definition of differential elements for a polar coordinate system is represented in Figure 22. With y D d A ds rd B C dr (, r ) Figure 22: Differential elements in a polar coordinate system reference to Figure 22, we define the differential surface as the area ABCD and compute the differential length as dl r = dr, dl θ = rdθ. We note here one important difference between the Cartesian differential lengths and dl θ and also between dl r and dl θ. We are interested in length, something that we could measure in meters. However, an infinitesimal change along the ˆθ direction has a magnitude dθ radian (or degree) and hence has units that are not in meters as any length. This represents the infinitesimal angle change going from the line defined as θ = θ 0 to the line defined as θ = θ 0 + dθ. To retrieve the length of the CD segment in Figure 22, we compute the arc length given by rdθ, that is the radius of the circle created by the arc segment times the angle in radians. Finally, the differential length is The differential surface area element is then dl = drˆr + rdθ ˆθ. ds = rdθdr. Later in this course we will determine the differential surface area for an arbitrary coordinate system using the concept of a Jacobian. 5 3D coordinate systems We now epand our space to three dimensions (3D) and present three common coordinate systems. The Cartesian and cylindrical will first be introduced as an etension to the previously described Cartesian and polar coordinate sets in 2D. Finally, we will conclude with the spherical coordinate system. In this section, we build on the previous section and several demonstrations will start from previously stated results. 24

25 5.1 Cartesian Definition In 3D, a position can be described by the intersection of three planar surfaces rather than by two straight lines. The Cartesian coordinate system is the most common coordinate set and we are used to referring to a position in terms of (, y, z). In this case, a position P ( 0, y 0, z 0 ) is described by three planes: = 0, y = y 0 and z = z 0 as shown in Figure 23 where 0 R, y 0 R and z 0 R. The intersection of three planes defines a point in 3D space. The and y part are identical to the 2D case and are respectively perpendicular to the yz and z plane. The base vectors ˆ and ŷ are also consistent with the 2D case. Now in 3D, we add a third dimension perpendicular to the plane formed by the y ais. This is the z dimension. z P ( o, yo, zo) ˆ ẑ ŷ y y o z z o y o o y Figure 23: 3D Cartesian coordinate system The three base vectors are unit vectors perpendicular to their respective surfaces. The base vectors are represented in Figure 23. When compared to the 2D case, we added ẑ perpendicular to both ˆ and ŷ. One thing to notice here again is the direction of the base vectors. The perpendicularity condition tells us that, according to Figure 23, ˆ and ˆ are valid base vectors along the ais and for ±ŷ and ±ẑ for the y ais and z ais respectively. There is an ambiguity in choosing a direction for the base vectors so we must impose a rule which we refer to as a right-handed system. There is also a left handed rule but it is infrequently used and must be stated up front. In a right handed system the base vectors satisfy the following rules: ˆ ŷ = ẑ, ŷ ẑ = ˆ, ẑ ˆ = ŷ. You can check by yourselves that the base vectors and aes in Figure 23 respect this condition by using the right-hand rule. Note, in sketching a problem you may rotate the aes to make drawing easier and you are free to label the aes as you wish but make sure that the rules above are satisfied. Then, any vector in 3D can be epressed in a Cartesian coordinate system as, v = v ˆ + v y ŷ + v z ẑ. 25