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Vector Algebra and Vector Calculus Chapter 2 Vector algebra and vector calculus Peter­Wolfgang Gr¨aber Systems Analysis in Water Management CHAPTER 2. VECTOR ALGEBRA AND VECTOR CALCULUS On the basis of simple, well-known representations of vector calculus the basic rules of the vector algebra are specified. Subsequently, the rules of vector differentiation with descriptive examples are discussed. ! 2.1 Unit vectors "ifferent unit vectors of vector representations are dependent on the use of coordinate system. #hen the vector �a can be expressed as the sum of multiples of the unit vectors. The unit vectors have the length $magnitude) one|e| & 1, and are always pa rallel to the coordinate system axis. (or the practical work in water management three coordinate systems, the Cartesian, the cylindri- cal and the spherical, are generally used. *n table !.' the vector �a is described in each of this three coordinate systems $also see figures !.! and !.'%. #able !.'+ coordinate systems for description of vectors coordinate unit vector�a system vectors → → → → → → → Cartesian i � j � k a&a x i,a y j,a z k → → → → → → → cylindrical r � φ� z a&a r r,a φ φ,a z z → → → → → → → sphericalrisch r � θ � φ a&a r r,a θ θ,a δ φ Figure 2.'+ vector representation in Cartesian coordinates In two-dimensional space the polar coordinate system will be used $see Figure 2.-%. Peter­Wolfgang Gr¨aber Systems Analysis in Water Management CHAPTER 2. VECTOR ALGEBRA AND VECTOR CALCULUS Figure 2.!+ vector representation in spherical coordinates Figure 2.-+ vector representation in two-dimensional space Since the vector �a is independent of the used coordinate system, the following conver- sion is applicable between the )artesian and the polar coordinate system+ 2 2 ar & ax ,a y &|�a| a a &� arctan y α a � x � $2.'% ax & cot $aα%·a y ax & cos $aα%·|�a| . 2.2 Algorithms In the following some important basic arithmetic rules for vectors are to be demon- strated by examples in the Cartesian coordinate system. • Addition The arguments of the )artesian unit vectors are respectively added in the vector addition+ �a, �b&$a x ,b x%�i,$a y ,b y%�j,$a z ,b z%�k $2.!% Notice: This relationship applies only to the )artesian coordinate system and can not be transferred to other coordinate systems. In the vector algebra the following laws apply: commutative law A�, B�& B�, A� $2.-% distributive lawm$n A�% & $mn%A�&n$m A�% $2..% distributive law$m,n% A�&m A�,n A� $!. % distributive lawm$ A�, B�%&m A�,m B� $!./% associative law A�,$ B�, C�%&$A�, B�%, C� $!.0% • Magnitude The magnitude of a vector is equal to its length and thus a scalar, which is direction-independent. 2 2 2 |�a|& ax ,a y ,a z $2.1% In paticular it applies that the magnitude� of the unit vector is equal to one. |�i|&| �j|&| �k|&|�r|&|�α|&| �δ|&' $ !.2% • Product There are two kinds of vector products with respect to the vector algebra, the scalar product $dot product% and the vector product$cross product). The scalar product between two vectors is defined+ �a· �b&|�a| · | �b| · cos(�a��b% $2.'3% Peter­Wolfgang Gr¨aber Systems Analysis in Water Management CHAPTER 2. VECTOR ALGEBRA AND VECTOR CALCULUS Hence the scalar product between two vectors is equal to 5ero, if they stand perpendicularly to each other. In particular it applies that the scalar product of a vector with itself, i.e. the square, is equal to the square of the magnitude: 3 �a⊥�b |�a| · |�b| �a� �b �a· �b& $2.''% −|�a| · |�b| �a�� �b |�a| · |�b| · cos �a��b generally � � 6articularly for the unit vectors: �i· �j & 37 �i· �k&37 �j· �k&37 �r· �α & 37 �r· �z & 37 $2.'!% �i· �i & '7 �j· �j & '7 �k· �k&'7 �r· �r&'7 �α· �α & '7 �z· �z & '7 According to the above algorithms we compute the scalar product as follows: �a· �b &$ax�i,a y�j,a z�k%·$b x�i,b y�j,b z�k% $2.'-% &a xbx ,a yby ,a zbz (or computing the angle between two vectors we use the equation+ a b ,a b ,a b cos(�a��b%& x x y y z z $2.'.% 2 2 2 2 2 2 ax ,a y ,a z bx ,b y ,b z � � The cross product between two vectors yields a vector+ �a× �b& �v $2.' % its magnitude is equal to the positive area of the parallelogram having �a and �b as sides |�v|&|�a× �b|&|�a| · | �b| · sin$�a��b% and its direction is perpendicularly to �a and �b+ �v⊥�a and �v⊥�b / 2.2. Algorithms Generally+ 3 �a��b |�a| · |�b| �a⊥�b |�a× �b|& $2.'/% −|�a| · |�b| �b⊥�a |�a| · |�b| · sin$�a��b% generally (or the Cartesian coordinate system applies: �i �j �k � � � � �a× �b& � � $2.'0% �ax ay az� � � � � � � �bx by bz � � � � � � � Especially for unit vectors: � � �i× �j &' �i× �k &' �j× �k &'|�r× �α|&'|�r× �z|&' � � � � � � � � �� � � � �� � � � �� � � i× j�& k � i× k�& j � j× k�& i �r× �α& �z �r× �z& �α �i× �i & 3 �j× �j & 3 �k× �k & 3|�r× �r|&3|�α× �α|&3|�z× �z|&3 � � � � � � $2.'1% � � � � � � � � � � � � Notice: (or the vector product the commutative law is not applicable, but the anticom- mutative+ �a× �b&− �b× �a $2.'2% The vector product is still distributive+ �a×$ �b, �c%& �a× �b, �a× �c $2.!3% • Differentiation In vector analysis we speak of three different kinds of differentiation, the gradient $grad%, the divergence $div% and the curl or rotation $rot% of a vector. (or all three methods a uniform differential vector, the Nabla operator∇ applies $see table !.!). #able !.- shows the ways of writing of the different kinds of differentiation in the overview as a function of the used coordinate system. (or 0 CHAPTER 2. VECTOR ALGEBRA AND VECTOR CALCULUS further simplification the Laplace differential operator� will be used. This is equal to the double application of the Nabla operator+ ;&∇·∇ $2.!'% #able !.!+ "escription of the <8=>8 operator in different coordinate systems coordinate system ∂ ∂ ∂ Cartesian �& �i, �j, �k ∂x ∂y ∂z ∂ ' ∂ ∂ cylindrical �& �r, · −→ϕ, �z ∂r r ∂ϕ ∂z ∂ ' ∂ ' ∂ −→ spherical �& �r, · −→ϕ, · θ ∂r r· sinθ ∂ϕ r ∂θ (or the gradient ∇ϕ & gradϕ $2.!!% scalar ϕ&⇒ vector∇ϕ ∂ −→ ∂ −→ ∂ −→ ∂ϕ−→ ∂ϕ−→ ∂ϕ−→ ∇ϕ& i, j, k ϕ& i, j, k ∂x ∂y ∂z ∂x ∂y ∂z � � the Nabla operator is applied to a scalar potential fieldϕ. The result for this is a vector. The gradient can be regarded as the formal multiplication of the Nabla operator with a scalar. In the field of the hydrogeology this quantity can be the groundwater levelh, temperature fieldsT , concentration distribu tionsC, evaporation or groundwater regeneration ratesv N and others. There scalars $potentials) are non- directional and have thereby no vector character. However they are location dependent. The most important application of the gradient is the Darcy law for the computation of the groundwater ?ow velocity (see section 0.' ). �v&−k gradh $2.!-% Example for the gradient: The groundwater level of an aquifer is indicated by the function+ h&!xy−-x,! 1 2.2. Algorithms @e compute the groundwater ?ow velocity for the case that the permeability coefficient of the aquifer isk&!·'3 −3m·s −1. It applies: �v&−k grad$h% ∂$!xy−-x,!% ∂$!xy−-x,!% ∂$!xy−-x,!% m &−!·'3 −3 �i, �j, �k ∂x ∂y ∂z s � � m m & $/−.y%'3 −3 ·�i−.·x·'3 −3 · �j s s It is to be recognized, that+ '. there is no vertical stream !. the velocity is dependent on the coordinates. Thus the current in the aquifer is not constant. The divergence is the application of the Nabla operator on a vector+ ∇�v &div �v $2.!.% vector �v&⇒ scalar ∂ ∂ ∂ ∂v ∂v ∂v ∇�v& �i, �j, �k v �i,v �j,v �k & x , y , z ∂x ∂y ∂z x y z ∂x ∂y ∂z � � � � The result of divergence formation is a scalar quantity. The divergence can be regarded as the formal application of the scalar product between the Nabla operator and a vector. According to the rule of scalar product formation the divergence of a vector is a scalar quantity. #he divergence, also noted as productivity of an area , indicates whether sources or sinks are in this area. *f the divergence of a vector field is equal to zero $∇�v &div �v & 3%, the area has neither a source nor a sink. According to GaussB law the entire source and sink activity of an area can be computed by the volume integral of the divergence. 8t the same time it is known from the balance laws that the difference between the source and sink activities, i.e. the ?ow rates, have to discharge through the surface+ div �v!"& �v· �n!#$ !.! % ���G �G � (or the two-dimensional area follows similarly+ div �v!A& �v· �n!$ $2.!/% ��A �L 2 CHAPTER 2. VECTOR ALGEBRA AND VECTOR CALCULUS �n is a normal (perpendicularly standing% unit vector to the surface or to the circumfer- ence. With GaussB theorem the volume integral can be converted into an integral over the surface and an area integral can be converted into an integral over the bound. Also the divergence plays a fundamental role in the hydrogeology, since all processes must be balance in the mathematical description. In particular a large number of further derivatives is based on the following relation+ div �v & div$−k gradh%&%$ !.!0% Example of divergence calculation: @e compute the divergence of the velocity vector �v for the previous example: ∂ ∂ ∇�v& $-−!y%!·'3 −3 , $−.·'3 −3x% & 3 ∂x ∂y Thus this area is neither source nor sink.
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