ChE 342 Vectors 1 VECTORS u u i k j u u Basic Definitions Figure 1 Vectors have magnitude and direction: u = i u + j u + k u (1) i = [1,0,0] j = [0,1,0] (2) k = [0,0,1] i, j, and k are mutuall orthogonal. Hence, b Pthagoras theorem, u 2 =u 2 +u 2 2 +u e 1 i ;e 2 j ;e 3 k u= e 1 u 1 +e 2 u 2 +e 3 u 3 = e i u e u i i i 3 i =1 The last two terms in Eq. (4) illustrate Einstein notation. (3) (4)
ChE 342 Vectors 2 Scalar Product u Figure 2 u.v= u.v cos (5) u.v= v.u Commutative Propert (6) u. ( v+w)=u.v +u.w Distributive Propert (7) u.v=u i e i.v j e j = u i v j e i.e j = u i v j δ ij = u i v i (8) δ ij = 1 i = j 0 i j Kronecker s delta (9) v Transformation of Coordinates r Figure 3 r = i + j+ k (10) = r.i = ( i+ j+k).i (11) j = i l ij [ i, j = 1,2,3; l ij = cosine (old ais i, new ais j)] (12)
ChE 342 Vectors 3 Vector Product u v =( u v sin)n uv (13) n uv is the unit vector normal to the plane containing u and v, pointing in the direction that a righthanded screw moves when turned from u to v. e i e j = ε ijk e k (14) u v v u Figure 4 1 123,231,312 ε ijk = 1 213,321,132 (15) 0 if 2 indices arethesame u v =u i e i v j e j =u i v j ε ijk e k = i( u u v )+ j( u v u )+ k( u v u v )= i j k (16) u u u v v
ChE 342 Vectors 4 Curvilinear Coordinates r r φ Clindrical Spherical Figure 5
# ( ' ' & ' # ChE 342 Tensors 1 TENSORS Second-order tensor: τ τ τ τ = τ τ τ (1) τ τ τ A second-order tensor is a linear relationship between two vectors, v τ τ τ u v =τ.u v = τ τ τ. u (2) τ T = τ τ τ τ τ τ τ τ τ τ τ τ u Transpose tensor (3) τ =τ T Smmetric tensor (4) τ = τ T Anti-smmetric tensor (5) τ = 1 ( 2 τ +τ T )+ 1 ( 2 τ τ T ) (6) uv = u v u v u u v u v u u v u v u Dadic product (7) δ = 1 0 0 0 1 0 0 0 1 Unit tensor (8) e 1 e 1 = 1 0 0 $ $ 0 0 0 $ ;e 1 e 2 = 0 1 0 0 0 0 $ $ $ ;etc. (9) 0 0 0 0 0 0 τ = e i e j τ ij (10) uv = e i e j u i v j (11) e i e j :e k e l =δ il δ jk e i e j.e k = e i δ jk e i.e j e k = e k δ ij e i e j.e k e l =e i e l δ jk ' Tensor products (12)
ChE 342 Differential Operators 1 VECTOR DIFFERENTIAL OPERATORS Gradient (Cartesian Coordinates) = i + j +k (1) Gradient of a scalar φ =i φ + j φ + k φ (2) Gradient of a vector v = i e i v j e j = v j i e i e j = v v v v v v (3) Divergence of a vector.v = e i.v j e j = v j δ i ij = v i = v i i + v + (4) Curl of a vector v = e i v j e j = v j e i e j = v j ε ijk e k = i i i i v + j v +k v v (5) Laplacian of a scalar. φ = e i. φ e i j = 2 φ δ j i ij = j 2 φ + 2 φ 2 + 2 φ 2 2 2 φ (6) Laplacian of a vector (.v) ( v) 2 v (7)
ChE 342 Differential Operators 2 EXPRESSING DIFFERENTIAL OPERATORS IN CURVILINEAR COORDINATES Clindrical Coordinates The relationship between cartesian and clindrical coordinates is given b, =rcos r= ( 2 + 2 ) 1 / 2 =rsin = arc tan( ) (8) = = Appling the chain rule, = r, r, cos r sin r Similarl, =sin r + cos r = + r,, +, r = (9) (10) (11) In addition, the relationship between cartesian and clindrical unit vectors is e r = cos i + sin j i =cos e r sin e e = sin i + cos j j = sin e r + cos e (12) e = k k =e
ChE 342 Differential Operators 3 Thus, the gradient operator becomes = i + j + k = cos r sin ( cos e r sin e )+ r sin r + cos ( sin e r +cos e )+ r k = r e + 1 r r e + e (13) Spherical Coordinates The analogous relations in spherical coordinates are: Transformation of coordinates, =rsin cosφ r= ( 2 + 2 + 2 ) 1 / 2 =rsin sinφ =arc tan( 2 + 2 ) (14) = rcos φ = arctan( ) Chain rule (onl shown) = r,φ r, + r,φ, + φ,r φ, (15) Relationship between unit vectors e r =sin cosφ i +sin sinφ j + cos k i = sin cosφ e r +cos cosφ e sinφ e φ e = cos cosφ i +cos sinφ j sin k j = sin sinφ e r + cos sinφ e + cosφ e φ (16) e φ = sinφi + cosφ j k =cos e r sin e And the gradient operator in spherical coordinates becomes, after analogous operations, = r e r + 1 r e 1 + r sin φ e φ (17)
ChE 342 Differential Operators 4 DIFFERENTIAL OPERATIONS IN CURVILINEAR COORDINATES To calculate an one of the differential operators, we proceed as follows. Consider the calculation of the divergence of a vector in clindrical coordinates,.v = e r r + e 1 r + e ( e r v r +e v + e )= e r. r e r v r +e r. r e v + e r. r e + e. 1 r e v + e.1 r r r e v + e.1 r e. e r v r +e. e v + e. e e + (18) Note that we need to know quantities such as e r. Let us compute them: e r = sin i +cos j = e e = cos i sin j = e r e = 0 e r r = 0 e r =0 e r = 0 (19) e r = 0 e =0 e =0 This then gives,.v = v r r + 1 v r + v r r + (20) The corresponding relations analogous to (19) in spherical coordinates are
ChE 342 Differential Operators 5 e r = e e = e r e φ =0 e r r = 0 e r =0 e φ r =0 (21) e r φ = sin e φ e φ = cos e φ e φ φ = sin e r cos e In general, then, using (13) and (19) (clindrical coordinates) or (17) and (21) (spherical coordinates), we can compute an differential operation. See, for eample, Tables A.7-1, A.7-2, A.7-3 in BSL. In performing these operations, products between vectors are interpreted in the usual wa. For eample, in clindrical coordinates, u.v =u r v r + u v + u (22) u v = e r ( u u v )+e ( u v r u r )+ e ( u r v u v r ) (23)