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Vector Calculus

For the velocity potential function, φ = x^2 - y^2 . Find the velocity components at the point (4, 5).

Vector Calculus

If for a two dimensional potential flow, the velocity potential is given by : φ = 4x(3y - 4), determine the velocity at the point (2, 3). Determine also the value of stream function ψ at the point (2, 3).

Vector Calculus

If F=A×(B×C) where A=5it^2 + (3t-2)j + 6tk; B=2i+4tj+3(1-t)k and C=4ti+5jt^2 -3tk evaluate integral of Fdt with limits zero to one.

Vector Calculus

What do you understand by unit tangent vector

Vector Calculus

Prove that the divergence of a curl is always zero:

∇⃗⃗. (∇⃗⃗

×a)=0

Vector Calculus

The position vector r of a particle at time t is

r = 2t

2

i + (t

2 − 4t)j + (3t − 5)k.

Find the velocity and the acceleration of the particle at time t. Show that when t =

2

5

the velocity

and the acceleration are perpendicular to each other. The velocity and the acceleration are resolved

into components along and perpendicular to the vector i − 3j + 2k. Find the velocity and acceleration

components parallel to this vector when t =

2

5

r = 2t

2

i + (t

2 − 4t)j + (3t − 5)k.

Find the velocity and the acceleration of the particle at time t. Show that when t =

2

5

the velocity

and the acceleration are perpendicular to each other. The velocity and the acceleration are resolved

into components along and perpendicular to the vector i − 3j + 2k. Find the velocity and acceleration

components parallel to this vector when t =

2

5

Vector Calculus

1. The position vector r of a missile at any time t is given by

r = ti + (8t − t

2

)j,

where i and j are unit vectors in the horizontal and upward vertical directions. Find its

(a) average velocity from t = 2 to t = 3,

(b) velocity, speed and direction of motion when t = 4,

(c) position vector when it is moving parallel to the vector i − 2j,

(d) acceleration vector.

2. The position vector r of a moving particle at time t after the start of the motion is given by

r = 5(1 + 4t)i + 5(19 + 2t − t

2

)j.

Find the initital velocity of the particle. At time t = T, the particle is moving at right angles to its initial

direction of motion. Find the value of T and the distance of the particle from its initial position at this

time.

r = ti + (8t − t

2

)j,

where i and j are unit vectors in the horizontal and upward vertical directions. Find its

(a) average velocity from t = 2 to t = 3,

(b) velocity, speed and direction of motion when t = 4,

(c) position vector when it is moving parallel to the vector i − 2j,

(d) acceleration vector.

2. The position vector r of a moving particle at time t after the start of the motion is given by

r = 5(1 + 4t)i + 5(19 + 2t − t

2

)j.

Find the initital velocity of the particle. At time t = T, the particle is moving at right angles to its initial

direction of motion. Find the value of T and the distance of the particle from its initial position at this

time.

Vector Calculus

Find the laplacian of the function (phi(x,y,z)=2x^2y-xzn^3

Vector Calculus

For a particle undergoing circular motion with an angular velocity w in in a circle of radius r show that

w×(w×r) = -w^2.r

w×(w×r) = -w^2.r

Vector Calculus

) For a scalar field ( , , ) ,

n n n

φ x y z = x + y + z show that (∇φ) r = nφ

r r

. , where n is a non-zero

real constant.

n n n

φ x y z = x + y + z show that (∇φ) r = nφ

r r

. , where n is a non-zero

real constant.