Vector Calculus: a quick review

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1 Appendi A Vector Calculus: a quick review Selected Reading H.M. Sche,. Div, Grad, Curl and all that: An informal Tet on Vector Calculus, W.W. Norton and Co., (1973). (Good phsical introduction to the subject) Mase, George. Theor and problems of Continuum Mechanics: Schaum s outline Series. (Heav on tensors but lots of worked problems) Marsden, J.E. and Tromba, A.J.. Vector Calculus. W.H. Freeman (or an standard tet on Vector calculus) In modeling we are generall concerned with how phsical properties change in space and time. Therefore we need a general mathematical description of both the variables of interest and their spatial and temporal variations. Vector calculus provides just that framework. A.1 Basic concepts Fields A field is a continuous function that returns a number (or sets of numbers) for ever point in space and time (, t). There are three basic flavours of fields we will deal with scalar fields A scalar field f(,,z,t) returns a single number for ever point in space and time. Eamples include temperature, salinit, porosit, densit.... vector fields A vector field F(,,z,t) returns a vector for ever point in space and is readil visualized as a field of arrows. Eamples include velocit, elastic displacements, electric or magnetic fields. tensor fields A second rank tensor field D(,,z,t) can be visualized as a field of ellipsoids (3 orthogonal vectors for ever point). Eamples include stress, strain, strain-rate. 1

2 2 Notation There are man different notations for scalars, vectors and tensors (and the re often mied and matched); however, a few of the common ones are scalars scalars are usuall shown in math italics e.g. f, g, t... vectors come in more flavours. when tpeset the re usuall bold-roman characters e.g. V, or the unit vectors i,j,k. When hand written the usuall have a line underneath them. Vectors can also be written in component form as v = v i + v j + v z k or in inde notation v = v i ê i where ê i is another representation of the unit vectors. tensors (actuall second rank tensors) Tpeset in sans-serif font D (or often just a bold σ), handwritten with two underbars, or b component D ij. 2nd rank tensors are also convenientl represented b matrices. Definitions of basic operations vector dot product a b = a i b i = a b cos θ (A.1.1) is a scalar that records the amount of vector a that lies in the direction of vector b (and vice versa). θ is the smallest angle between the two vectors. vector cross product c = a b is a vector that is perpendicular to the plane spanned b vectors a and b. The direction that c points in is determined b the right hand rule. Note a b = b a. The cross product is most easil calculated as the determinant of the matri or c = i j k a a a z b b b z (A.1.2) c = (a b z a z b )i (a b z a z b )j + (a b a b )k (A.1.3) or in inde notation as c i = ɛ ijk a j b k where ɛ ijk is the horrid permutation smbol. tensor vector dot product is a vector formed b matri multiplication of a tensor and a vector c = D a. In the case of stress, the force acting on a plane with normal vector n is simpl f = σ n. Each component of the vector is most easil calculated in inde notation with c i = D ij a j with summation implied over repeated indices (i.e. c 1 = D 11 a 1 + D 12 a 2 + D 13 a 3 and so on for i = 2,3.

3 Vector calculus review 3 A.2 Partial derivatives and vector operators Definitions Given a scalar function of one variable f(), its derivative is defined as df d = lim f( + ) f() (A.2.1) 0 (and is locall the slope of the function). Given a function of more than one variable, f(,, t), the partial derivative with respect to is defined as f = lim f( +,,t) f(,,t) 0 (A.2.2) i.e. if we sliced the function with a plane ling along, the partial derivative would be the slope of the function in the direction of (See Figure A.1a) likewise or t. In space in fact it is convenient to consider all of the spatial partial derivatives together in one hand package, the del operator ( ) a.k.a. the upside down triangle.. In Cartesian coordinates this operator is defined as = i + j + k z (A.2.3) In combination with vector and scalar fields, the del operator gives us important information on how these fields var in space. In particular, there are 3 important combinations the Gradient the gradient of a scalar function f() f = i f + j f + k f z (A.2.4) is a vector field where each vector points uphill in the direction of fastest increase of the function (See Figure A.2). the Divergence the divergence of a vector field F = F + F + F z z (A.2.5) is a scalar field that describes the strength of local sources and sinks. If F = 0 the field has no sources or sinks and is said to be incompressible. the Laplacian the Laplacian of a scalar field 2 f = ( f) = 2 f f f z 2 is a scalar field that gives the local curvature (See Figure A.3). (A.2.6) the Curl the curl of a vector field F = ( F z F z )i ( F z F z )j + ( F F )k is a vector field that that describes the local rate of rotation or shear. (A.2.7)

4 4 Other useful relationships Given the basic definitions, there are several identities and relationships that will be important for the derivation of conservation equations. Gauss divergence theorem Gauss s theorem states that the flu out of a closed surface is equal to the sum of the divergence of that flu over the interior of that volume (it is actuall closel related to the definition of the Divergence). Mathematicall F ds = FdV (A.2.8) useful identities the first homework will make ou show that S 1. ( F) = 0 (i.e. if a vector field can be written as V = g then it is automaticall incompressible). 2. ( f) = 0 (a gradient field is irrotational) 3. V = ( V) 2 V 4. [(V )V] = (V )[ V] Figures A.1 A.5 show some eamples of scalar and vector fields and their derivatives. V -1 f a -1 1 b Figure A.1: (a) Surface plot of the 2-D scalar function f(, ) = 5 ep [ ( 2 / /25) ] (b) contour plot of the same function.

5 Vector calculus review = 0.86 Figure A.2: Vector plot of f(, ) for the function f in Figure A Figure A.3: contour plot of of the divergence of the vector field in Figure A.2. Because ( f) = 2 f this plot is also a measure of the curvature of the function f.

6 a = 1.00 b Figure A.4: (a) Vector plot of the 2-D corner flow velocit field V(, ) = 2/π [( tan 1 (/) /( ) ) i 2 /( )j ] (b) contour plot of log 10 ( V k). The maimum rate of rotation is in the corner. There is no rotation directl on the ais. This field is incompressible however and V = = Figure A.5: Vector plot of pure-shear flow field V(, ) = i j. Although the flow lines of this field are superficiall similar to those of Figure A.4, this flow is locall irrotational i.e. V = 0. This flow is also incompressible.