problems on PARTIAL DIFFERENTIATION of vector functions : Vector calculus #problems #on #partial #derivative #of #vector #functions : #Vector #calculus

problems on partial differential of vector functions : Vector Calculus:

Problem :

If f = ( 2x²y-x⁴ ) i + ( e^xy – ysinx ) j + (x² cosy ) k then find ∂²f/∂x² and ∂²f/∂x∂y.

Solution :

Problem : If A = 2x² i -3yz j +xz² k and ⲫ = 2z-x³y then find

A. [ i ∂ⲫ /∂x + j ∂ⲫ /∂y + k ∂ⲫ /∂z ]

and A x [ i ∂ⲫ /∂x + j ∂ⲫ /∂y + k ∂ⲫ /∂z ] at (1,-1,1)

Solution :

Problem :

If ⲫ = x²yx+4xz² and A= 2 i -j -2k then find A. [ i ∂ⲫ /∂x + j ∂ⲫ /∂y + k ∂ⲫ /∂z ] at ( 1,-2,1).

Solution :

Problem :

If ⲫ = 2xz⁴ -x² y then find the value of | i ∂ⲫ /∂x + j ∂ⲫ /∂y + k ∂ⲫ /∂z | at ( 2,-2,1).

Solution :

Problem :

If r = x i + y j +z k and ⲫ = log r , prove that

i ∂ⲫ /∂x + j ∂ⲫ /∂y + k ∂ⲫ /∂z = r / r²

^{Solution :}

Problem :

If f = yz i + zx j + xy k then prove that [ i x ∂f /∂x + j x ∂f /∂y + k x ∂f /∂z ] = 0.

Solution :

Problem : If A = cosxy i + (3xy-2x² ) j – ( 3x+2y ) k then find ∂²f/∂y² and ∂²f/∂x∂y.

Solution :

Problem :

If r = x i + y j +z k and a is constant vector prove that

i ∂(a.r) /∂x + j ∂(a.r) /∂y + k ∂(a.r) /∂z = a.

Solution :

Problem :

If r = x i + y j +z k and a is constant vector prove that

∂ /∂x ( (a x r) x i ) + ∂ /∂y ( ( a x r ) x j ) + ∂ /∂z ( ( a x r ) x k ) = -2a

Solution :

#problems #on #partial #derivative #of #vector #functions : #Vector #calculus

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