Differentiation
i, j, k are
Unit vectors
Gradient of Φ
∇Φ
Directional derivative of Φ is
e.∇Φ
Divergence of a vector is the dot product between
∇ and the vector
Curl of a vector is this product between ∇ and the vector
Cross product
The derivative of the position vector with respect to time gives this vector.
Velocity vector
Maximum Rate of Change
|∇Φ| denotes
Maximum value of directional derivative
|∇Φ|
A vector is named this if it's divergence vanishes
Solenoidal vector
The vector is named this if it's Curl vanishes
Irrotational vector
Two vectors are not this if their dot product is nonzero.
Orthogonal vectors
Φ=x2yz , then GradΦ at (1, 1, 1) is
2i+j+k
n1.n2=0
Two normal vectors are orthogonal
This measures how much a vector field spreads out from a point is this
Divergence
Φ is this if a vector f is represented as ∇Φ
Scalar Potential
The derivative of a constant vector with respect to any variable is this
Zero vector
If a=i+2j+3k, then unit vector along a is
e = ( 1/√14 ) i+2j+3k
i, j, k are unit vectors along
X, Y and Z axis respectively
if f=2i+3j-4k, then div f=
0
This determines the rotation or twist of a vector field
Curl
This operation finds the rate of change of a vector function with respect to a scalar variable.
What is vector differentiation?
Unit Normal Vector to the surface Φ =x2+y2+z2
(1/√3) (xi+yj+zk)
Directional derivative of a scalar potential is calculated in the direction of
A vector
if f=xi+2y2j-4zk, then div f=
4y-3
If f=grad(x3+y3+z3), then curl f =
0