Mathematical Methods in Fluid Dynamics Scalars and Vectors Scalar – any quantity which can be fully specified by a single number Vector – a quantity which requires both a magnitude and direction to be fully specified What are some examples of scalar and vector quantities? Vector Notation: €  u = ux i + uy j Magnitude of a vector: €  u = ux2+ uy2 Direction of a vector: € Direction = arctan uyux( ) Coordinate systems on the Earth: For a coordinate system with (x,y,z) we use unit vectors €  i , €  j , and €  k .Algebra of Vectors Addition and subtraction of two vectors (graphic method): Addition of two vectors €  u and €  v : €  u = ux i + uy j + uz k and v = vx i + vy j + vz k ⇒ u + v = (ux+ vx) i + (uy+ vy) j + (uz+ vz) k Subtraction of two vectors €  u and €  v : €  u − v = (ux− vx) i + (uy− vy) j + (uz− vz) k Multiplication of a vector by a scalar (graphic method):Multiplication of vector €  u by scalar c: € c u = cux i + cuy j + cuz k How does the direction and magnitude of a vector change due to multiplication by a scalar? Multiplication of two vectors Scalar product (or dot product) of €  u and €  v : €  u • v = uxvx+ uyvy+ uzvz= u  v cosθ When will the dot product of two vectors be equal to zero? What does this tell us about the direction of the two vectors relative to each other?Vector product (or cross product) of €  u and €  v : u ×v = (uyvz− uzvy)i + (uzvx− uxvz)j + (uxvy− uyvx)k or u ×v =ijkuxuyuzvxvyvz= (uyvz− uzvy)i + (uzvx− uxvz)j + (uxvy− uyvx)k What is the direction of the vector that results from the cross product? The right hand rule What is the magnitude of this vector? Magnitude = €  u  v sinθ When will the cross product be equal to zero?Scalar and Vector Fields Field – a quantity defined over a given coordinate space The field is a function of the three coordinates of position and also of time. T = f(x,y,z,t) T and p are examples of scalar fields and wind is an example of a vector field. Coordinate Systems on the Earth How do scalar and vector fields change when the coordinate system is rotated? How would vector €  u = ux i + uy j change under a rotation of the coordinate system? €  " u = (uxcosα+ uysinα) " i + (uycosα− uxsinα) " j Meteorologists traditionally define a coordinate system relative to the Earth.Gradients of Vectors The vectors we consider in meteorology often vary in space and time (i.e. they are functions of both space and time). We can show this, for a wind velocity vector, as €  u x, y,z,t( ). This vector, in component form, can be written as: €  u x, y,z,t( )= f (x,y,z,t) i + g(x,y,z,t) j + h(x,y,z,t) k Written in this way we see that this vector consists of zonal, meridional, and vertical components of the wind that vary in all three spatial directions (x,y,z) and vary in time. The variation of the wind vector with respect to any one of the independent variables can be written as a partial derivative. € ∂ u ∂t=∂f∂t i +∂g∂t j +∂h∂t k What does each term in this equation represent physically? What if we considered the partial derivative of €  u (x,y,z,t) = f (x,y,z,t) i + 0 j + 0 k with respect to x or