Mathematical Methods for Physics and Engineering : A Comprehensive Guide

10.7 VECTOR OPERATORS x y z nˆ 0 (0, 0 ,a) O a φ=x^2 +y^2 +z^2 =a^2 z=a Figure 10.6 The tangent plane and the normal to the surf ...

VECTOR CALCULUS In addition to these, we note that the gradient operation also obeys the chain rule as in ordinary differential ...

10.7 VECTOR OPERATORS 10.7.3 Curl of a vector field Thecurlof a vector fielda(x, y, z) is defined by curla=∇×a= ( ∂az ∂y − ∂ay ∂ ...

VECTOR CALCULUS ∇(φ+ψ)=∇φ+∇ψ ∇·(a+b)=∇·a+∇·b ∇×(a+b)=∇×a+∇×b ∇(φψ)=φ∇ψ+ψ∇φ ∇(a·b)=a×(∇×b)+b×(∇×a)+(a·∇)b+(b·∇)a ∇·(φa)=φ∇·a+a·∇φ ...

10.8 VECTOR OPERATOR FORMULAE
Show that ∇×(φa)=∇φ×a+φ∇×a. Thex-component of the LHS is ∂ ∂y (φaz)− ∂ ∂z (φay)=φ ∂az ∂y + ∂φ ∂y ...

VECTOR CALCULUS ais a vector field, these four combinations are grad(gradφ), div(diva), curl(diva) and grad(curla). In each case ...

10.9 CYLINDRICAL AND SPHERICAL POLAR COORDINATES (ii) If the unit vectors vary as the values of the coordinates change (i.e. are ...

VECTOR CALCULUS ρ, φ, z,where x=ρcosφ, y=ρsinφ, z=z, (10.44) andρ≥0, 0≤φ< 2 πand−∞<z<∞. The position vector ofPmay ther ...

10.9 CYLINDRICAL AND SPHERICAL POLAR COORDINATES x y z z ρ r i j k O P eˆz ˆeφ ˆeρ φ Figure 10.7 Cylindrical polar coordinatesρ, ...

VECTOR CALCULUS ∇Φ= ∂Φ ∂ρ ˆeρ+ 1 ρ ∂Φ ∂φ ˆeφ+ ∂Φ ∂z ˆez ∇·a = 1 ρ ∂ ∂ρ (ρaρ)+ 1 ρ ∂aφ ∂φ + ∂az ∂z ∇×a = 1 ρ ∣ ∣∣ ∣ ∣∣ ∣ ∣ eˆρ ρe ...

10.9 CYLINDRICAL AND SPHERICAL POLAR COORDINATES x y z r i j k O θ P ˆer ˆeφ ˆeθ φ Figure 10.9 Spherical polar coordinatesr, θ, ...

VECTOR CALCULUS andr≥0, 0≤θ≤πand 0≤φ< 2 π. The position vector ofPmay therefore be written as r=rsinθcosφi+rsinθsinφj+rcosθk. ...

10.9 CYLINDRICAL AND SPHERICAL POLAR COORDINATES ∇Φ= ∂Φ ∂r ˆer+ 1 r ∂Φ ∂θ eˆθ+ 1 rsinθ ∂Φ ∂φ ˆeφ ∇·a = 1 r^2 ∂ ∂r (r^2 ar)+ 1 rs ...

VECTOR CALCULUS 10.10 General curvilinear coordinates As indicated earlier, the contents of this section are more formal and tec ...

10.10 GENERAL CURVILINEAR COORDINATES z x i y j k O P u 2 =c 2 u 1 =c 1 u 3 =c 3 u 1 u 2 u 3 ˆ 1 ˆ 2 ˆ 3 ˆe 1 eˆ 2 ˆe 3 Figur ...

VECTOR CALCULUS For orthogonal coordinates this is given by dV=|du 1 e 1 ·(du 2 e 2 ×du 3 e 3 )| =|h 1 eˆ 1 ·(h 2 eˆ 2 ×h 3 ˆe 3 ...

10.10 GENERAL CURVILINEAR COORDINATES In the last step we have used the chain rule for partial differentiation. Thereforeei·j=1 ...

VECTOR CALCULUS
Prove the expression for∇·ain orthogonal curvilinear coordinates. Let us consider the sub-expression∇·(a 1 ˆe 1 ...

10.11 EXERCISES ∇Φ= 1 h 1 ∂Φ ∂u 1 ˆe 1 + 1 h 2 ∂Φ ∂u 2 ˆe 2 + 1 h 3 ∂Φ ∂u 3 ˆe 3 ∇·a = 1 h 1 h 2 h 3 [ ∂ ∂u 1 (h 2 h 3 a 1 )+ ∂ ...

VECTOR CALCULUS 10.3 The general equation of motion of a (non-relativistic) particle of massmand chargeqwhen it is placed in a r ...