University:
California State University, Northridge
Course:
PHYS 100B | General Physics II
Academic year:

2008
Views:

206

Pages:

11
Author:

Rylan Edwards

Review: Vector Algebra And Analysis R1.1 R1.2 R1.3 R1.4 R1.5 R1.6 R1.7 Scalar Product Vector Product Gradient and Del operators Divergence Curl of a Vector Field Laplacian Operator Other Operators Scalar and Vector Products For two vectors: G G G G a=a x i + a y j + az k G G G G b=bx i + by j + bz k θ G G and θ is the angle between a and b. G G The magnitudes of a and b are: a= a 2x + a y2 + az2 , b= bx2 + by2 + bz2 , R1.1 Scalar product G G a ⋅ b=abcosθ G G a ⋅ b=a x bx + a y by + az bz (1.1) (1.2) G b G a R1.2 Vector product G G a × b =absinθ G G G G a × b= - b × a (2.1) (2.1) G G The direction of a × b is determined by the Right-Hande-Rule. G G G G G G G G a × b=(a x i + a y j + az k ) × (bx i + by j + bz k) G G G =(a ybz -a z b y ) i + (a z bx -a x b z )j+ (a x by -a y b x )k (2.3) The above equation can also be expressed as : i G G a × b = ax bx j k ay by az bz (2.4) 1.3 Gradient (Grad) and del Operators Fig.3 Two close contours in a scalar field in the x-y coordinate system. Gradient (Grad) and del Operators ∇ is a Vector Operator: ∂ G ∂ G ∂ G i+ j+ k ∇= ∂x ∂y ∂z ∂ G ∂ G ∂ G Grad φ = ∇φ =( i+ j+ k)φ ∂x ∂y ∂z ∂φ G ∂φ G ∂φ G = i+ j+ k ∂x ∂y ∂z The gradient is the maximum rate of change of the scale field φ , which is perpendicular to the contour at that point. Physical Example: Electric potential and electric field. 1.4 Divergence (Div) G Fig. 5. The relationship of V and the G small vector area A Fig. 6. The cube box surrounding the G point P, and the vector field V. Divergence (Div) G G G G G ∂ G ∂ G ∂ G i+ j+ k) ⋅ (Vx i + Vy j + Vz k ) Div V = ∇ ⋅ V =( ∂x ∂y ∂z ∂Vx ∂Vy ∂Vz = + + (4.6) ∂x ∂x ∂x The divergence is the outward flux per unit volume at that point. It is a scalar quantity. Physical Example: electric flux per unit volume. Divergence Theorem v∫ Sv G G G V ⋅ d s= ∫ (∇ ⋅ V )dv V 1.5 Curl of a Vector Field Fig. 6. The circulation in a square contour in the xy plane. Curl of a Vector Field G i G j G k G G ∂ Curl V = ∇ × V = ∂x Vx ∂ ∂y Vy ∂ ∂z Vz ∂Vz ∂Vy G ∂Vx ∂Vz G ∂Vy ∂Vx G =( )i+( )j+( )k ∂y ∂z ∂z ∂x ∂x ∂y G The Z components of curl V is the circulation ∫ c (5.2) G G V ⋅ ds per unit area in the x-y plane. G In a x-y-z coordinate, the vector field V has a rotatory component in a plane whose G normal is in the direction of V. The circulation per unit area in that plane is given by G ∇× V 1.6 Laplacian Operator Laplacian operator is a scalar operator, and is defined as : ∇ 2 = ∇ ⋅∇ ∂2 ∂2 ∂2 = 2+ 2+ 2 ∂x ∂y ∂z (6.1) 1.7 Other operators G G G G For a vector field V = Vx i + Vy j + Vz k , we can prove that G G G 2 ∇ × ∇ × V = ∇(∇ ⋅ V ) − ∇ V (7.1)