Gradient, Divergence and Curl
We will cover following topics
Vector Differential Operator
The symbolic vector →i∂∂x+→j∂∂y+→k∂∂z is called Hamiltonian operator or vector differential operator and is denoted by ∇ (read as del or nabla).
∴∇=→i∂∂x+→j∂∂y+→k∂∂zIt is also known as del operator. This operator can be applied on a scalar point function ϕ(x,y,z) or a vector point function →F(x,y,z) which are differentiable functions. This gives rise to three field quantities namely gradient of a scalar, divergence of a vector and curl of a vector function.
Gradient
If ϕ(x,y,z) is a scalar point function continuously differentiable in a given region R of space, then the gradient of ϕ is defined by ∇ϕ=→i∂ϕ∂x+→j∂ϕ∂y+→k∂ϕ∂z.
∇ϕ is normal to the surface ϕ(x,y,z)=C at P. A unit normal to the surface at P is →n=∇ϕ|∇ϕ|.
Directional Derivative
The directional derivative of a scalar point function ϕ in a given direction →a is the rate of change of ϕ in that direction. It is given by the component of ∇ϕ in the direction of →a.
∴ The directional derivative =∇ϕ⋅→a|→a| since ∇ϕ⋅→a|→a|=|∇ϕ||→a||→a|cosθ, where θ is the angle between ∇ϕ and →a
Since ∇ϕ⋅→a|→a|=|∇ϕ||→a||→a|cosθ, where θ is the angle between ∇ϕ and →a =|∇ϕ|cosθ
So, the directional derivative at a given point is maximum if cosθ is maximum. i.e., cosθ=1⇒θ=0 ∴ the maximum directional derivative at a point is in the direction of ∇ϕ and the maximum directional derivative is |∇ϕ|
Equation of Tangent Plane
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The vector equation of the tangent plane at the point A is (→r−→r0)⋅∇ϕ=0.
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The cartesian equation of the plane at the point A(x0,y0,z0) is
where the partial derivatives are evaluated at the point (x0,y0,z0).
Equation of Normal Plane
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The vector equation of the normal at the point A is (→r−→r0)×∇ϕ=0.
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The cartesian equation of the normal at the point A is
where the partial derivatives are evaluated at (x0,y0,z0).
Angle between Two Surfaces at a Common Point
The angle between two surfaces f(x,y,z)=C1 and g(x,y,z)=C2 at a common point P is the angle between their normals at the point P.
| If θ is the angle between the normals at the point P, then $$\cos \boldsymbol{\theta}=\dfrac{\nabla f \cdot \nabla g}{ | \nabla f | \nabla g | }$$. |
| If θ=π2, then the normals are perpendicular and $$\cos \boldsymbol{\theta}=0 \Rightarrow \dfrac{\nabla f \cdot \nabla g}{ | \nabla f | \nabla g | }=0 \Rightarrow \nabla f \cdot \nabla g=0$$. |
∴ if two surfaces are orthogonal at the point P then ∇f⋅∇g=0.
Divergence
If →F(x,y,z) be a vector point function continuously differentiable in a region R of space, then the divergence of →F is defined by
∇⋅→F=→i⋅∂→F∂x+→j⋅∂→F∂y+→k⋅∂→F∂zIt is abbreviated as div→F and thus, div→F=∇⋅→F
If →F=F1→i+F2→j+F3→k, then ∇⋅→F=∂F1∂x+∂F2∂y+∂F3∂z
If →F is a constant vector, then ∇⋅→F=0 and conversely if ∇⋅→F=0, then →F is a constant vector.
Physical interpretation of divergence applied to a vector field is that it gives approximately the ‘loss’ of the physical quantity at a given point per unit volume per unit time.
Curl
If →F(x,y,z) be a vector point function continuously differentiable in a region R, then the curl of →F is defined by
∇×→F=→i×∂→F∂x+→j×∂→F∂y+→k×∂→F∂zIt is abbreviated as curl ˉF Thus, curl→F=∇×→F If →F=F1→i+F2→j+F3→k , then curl →F=∇×→F=(→i∂∂x+→j∂∂y+→k∂∂z)×(F1→i+F2→j+F3→k)=→i[∂F3∂y−∂F2∂z]+→j[∂F1∂z−∂F3∂x]+→k[∂F2∂x−∂F1∂y]
This is symbolically written as ∇×→F=|ccc→i→j→k∂∂x∂∂y∂∂z F1 F2 F3|.
PYQs
Gradient
1) Find the directional derivative of the function xy2+yz2+zx2 along the tangent to the curve x=t, y=t2, z=t3 at the point (1,1,1).
[2019, 10M]
2) Find f(r) such that ∇f=→rr5 and f(1)=0.
[2016, 10M]
3) Examine whether the vectors ∇u, ∇u and ∇w are coplanar, where u, v and w are the scalar functions defined by:
u=x+y+zv=x2+y2+z2 and w=yz+zx+xy.
[2011, 15M]
4) Show that the vector field defined by the vector function →v=xyz(yzˆi+xzˆj+xyˆk) is conservative.
[2010, 12M]
5) Find the directional derivative of f(x,y)=x2y3+xy at the point (2,1) in the direction of a unit vector which makes an angle of π3 with the x−axis.
[2010, 10M]
6) Find the directional derivative of
i) 4xz3−3x2y2z2 at (2,−1,2) along z−axis
ii) −x2yz+4xz2 at (1,−2,1) in the direction of 2ˆi−ˆj−2ˆk.
[2009, 12M]
7) If →r denotes the position vector of a point and if ˆr be the unit vector in the direction of →r, r=|→r|, determine grad (r−1) in terms of ˆr and r.
[2007, 12M]
8) Find the values of constants a, b and c so that the directional derivative of the function f=axy2+byz+cz2x2 at the point (1,2,−1) has maximum magnitude 64 in the direction parallel to z−axis.
[2006, 12M]
9) Find the values of constants a, b and c such that the maximum value of directional derivative of f=axy2+byz+cx2z2 at (1,−1,1) is in the direction parallel to y−axis and has magnitude 6.
[2002, 15M]
10) Find the directional derivative of f=x2yz3 along x=e−t, y=1+2sint, z=t−cost at t=0.
[2001, 15M]
Divergence
1) For what values of the constant a, b and c the vector ¯V=(x+y+az)ˆi+(bx+2y−z)ˆj+(−x+cy+2z)ˆk is irrotational. Find the divergence in cylindrical coordinates of the vector with these values.
[2017, 10M]
2) Prove that the divergence of a vector field is invariant with respect to co-ordinate transformations.
[2003, 12M]
Curl
1) Find the circulation of →F round the curve C, where →F=(2x+y2)ˆi+(3y−4x)ˆj and C is the curve y=x2 from (0,0) to (1,1) and the curve y2=x from (1,1) to (0,0).
[2019, 15M]
2) A vector field is given by →F=(x2+xy2)ˆi+(y2+x2y)ˆj. Verify that the field is irrotational or not. Find the scalar potential.
[2015, 12M]
3) If →r be the position vector of a point, find the value(s) of n for which the vector rn→r is:
i) irrotational,
ii) solenoidal.
[2011, 15M]
4) Show that →F=(2xy+z3)ˆi+x2ˆj+3xz2ˆk is a conservative force field. Find the scalar potential for →F and the work done in moving an object in this field from (1−2,1) to (3,1,4).
[2008, 12M]
5) For any constant vector, show that the vector →a represented by curl (→a×→r) is always parallel to the vector →a, →r being the position vector of a point (x,y,z) measured from the origin.
[2007, 15M]
6) If →r=xˆi+yˆj+xˆk, find the value(s) of in order that rn→r may be:
i) solenoidal
ii) irrotational.
[2007, 15M]
7) Prove that rn¯r is an irrotational vector for any value of n but is solenoidal only if n+3=0.
[2006, 15M]
8) Prove that the curl of a vector field is independent of the choice of coordinates.
[2005, 12M]
9) Show that if →A and →B are irrotational, then →A×→B is solenoidal.
[2004, 12M]
10) Show that the vector field defined by
→F=2xyz3\vecfi+x2z3→j+3x2yz2→kis irrotational. Find also the scalar u such that F=grad u.
[2001, 15M]