Problem Set 2Advanced Engineering Mathematics (ME 501)

Department of Mechanical EngineeringIndian Institute of Technology Guwahati

1. Show that the torsionτ of a curveC : r(t) is (κ > 0)

τ(s) = (u p p′) or (T N N′

) = (r′ r′′ r′′′)/κ2 whereφ′ =dφ

ds

τ(t) =(r′ × r′′) · r′′′

(r′ · r′)(r′′ · r′′)− (r′ · r′′)2whereφ′ =

dφ

dt

note(a1 a2 a3) implies triple product of vectorsa1, a2 anda3.

2.(a) Find the distance between the point (2,2,0) and the linex+ y = 0, y − z = 1.

(b) The lineL, whose equations arex−2z−3 = 0, y−2z = 0 intersects the planex+3y−z+4 = 0.Find the point of intersectionP and find the equation of that line in this plane that passes throughP and is perpendicular toL.

(c) Find the equation of the plane that passes through the points(−2, 0,−3) and(1,−2, 1) and isparallel to the line joining the points(−2,−13/5, 26/5) and(16/5,−13/5, 0).

3. Draw the following curves and calculate their curvature

r(t) = (a cos t, b sin t)

r(t) = (a cos t, a sin t, ct)

r(t) = (t, t3/2)

r(t) = (cosh t, sinh t)

xy = c

y = x2.

4. Find the values of the constantsa, b, c such that the maximum value of the directional derivativeof f(x, y, z) = axy2 + byz + cx2z2 at (1,−1, 1) is in the direction parallel to the axis ofy and hasmagnitude 6.

5. Evaluate the following line integrals

f = x2 + y2 + z2, C : (cos t, sin t, 2t), 0 ≤ t ≤ 4π

f =√

2 + x2 + 3y2, C : (t, t, t2), 0 ≤ t ≤ 3

f = 1− sinh2 x, C : (t, cosh t), 0 ≤ t ≤ 2

f = x2 + (xy)1/3, C : (cos3 t, sin3 t), 0 ≤ t ≤ π

f =√

16x2 + 81y2, C : (3 cos t, 2 sin t), 0 ≤ t ≤ π

6. Prove the following identities assuming continuous firstpartial derivative

a× (b× c) + b× (c× a) + c× (a× b) = 0.

∇ · (f∇g × g∇f) = 0.

∇ · (∇f ×∇g) = 0.

∇× (a× b) = (b · ∇)a− (a · ∇)b+ (∇ · b)a− (∇ · a)b.

∇ · (a× b) = −a · (∇× b) wherea is a constant vector.

∇× (a× r) = 2a wherea is a constant vector.

∇2(fg) = g∇2f + 2∇f · ∇g + f∇2g wheref, g are smooth functions.

If ∇ · a = 0 then∇× (a · ∇a) = a · ∇(∇× a)− (∇× a) · ∇a.

7. Check for path independence. In case of independence integrate from(0, 0, 0) to (a, b, c).

2xy2dx+ 2x2ydy + dz

ezdx+ 2ydy + xezdz

cos(x+ yz)(dx+ zdy + ydz)

3(x+ y)2(dx+ 2dy) + dz

−2xdx+ z sinh y dy + cosh y dz

8. Evaluate the following line integral∮

CF · dr counterclockwise around the boundaryC of the

regionR

F = (tan(x/5), x5y), R : x2 + y2 ≤ 25, y ≥ 0

F = (x cosh 2y, 2x2 sinh 2y), R : x2≤ y ≤ x

F = (ey/x, ey ln x+ 2x), R : 1 + x4≤ y ≤ 2

9. Show that a regionT with boundary surfaceS has the volume

V =1

3

∫

S

∫

r cosφ dA

where r is the distance of a variable pointP : (x, y, z) onS from the originO andφ is the anglebetween the directed lineOP and the outer normal ofS atP .

10. Verify the Divergence theorem for (A)f = 2z2 − x2 − y2 andS the surface of the box0 ≤ x ≤ 1, 0 ≤ y ≤ 2, 0 ≤ z ≤ 4 and (B)f = x2 − y2 andS the surface of the cylinderx2 + y2 ≤ 4, 0 ≤ z ≤ 1.

11. Evaluate the line integral∫

CF · r′(s)ds in right-handed Cartesian coordinate system

F = (−5y, 4x, z) C : x2 + y2 = 4, z = 1

F = (y, z/2, 3y/2) C : x2 + y2 + z2 = 6z, z = x+ 3

F = (2y2, x,−z3) C : x2 + y2 = a2, z = b(> 0)