Indian Institute of Space Science and Technology
Thiruvananthapuram
Vector Calculus Tutorial
Directional Derivative & Arc Length Function
1. Consider f:R2−→ Rdefined by f(x, y) := ||x|−|y|| − |x|−|y|. Determine whether (i) fis
continuous at (0,0), (ii) the partial derivatives Dxf|(0,0) and Dyf|(0,0) exist, and (iii) the directional
derivative D~vf|(0,0) exists. Is fdifferentiable at (0,0)? Justify your answer.
2. Let f:R2−→ Rbe defined by f(x, y) := 0 if xy = 0, and f(x, y) := 1 otherwise. Show that fis
not continuous at (0,0) although both the partial derivatives of fexist at (0,0).
3. Let f:R2−→ Rbe defined by f(x, y) := x2+y2if xand yare both rational, and f(x, y) := 0
otherwise. Determine the points of R2at which (i) Dxfexists, (ii) Dyfexists.
4. Let f:R2−→ Rdefined by one of the following functions. Check if D~v f|(0,0) exists for any unit
vector ~v. Is fcontinuous at (0,0)? Is fdifferentiable at (0,0)?
(i) f(x, y) = px2+y2, (ii) f(x, y) = |x|+|y|
5. Consider f:R2−→ Rdefined by f(0,0) := 0 and for (x, y)6= (0,0), by one of the following. In
each case, determine whether the directional derivative D~vf|(0,0) exists for any unit vector ~v in R2.
If it does, then check whether D~v f|(0,0) =h∇f(0,0), ~v ifor a unit vector ~v in R2. Finally, determine
whether fis differentiable at (0,0).
(i) x2y
x2+y2, (ii) xy x2−y2
x2+y2, (iii) x3
x2+y2, (iv) xy2
x4+y2, (v) ln(x2+y2), (vi) xyln(x2+y2), (vii) xy
x2+y2.
6. Consider f:R2−→ Rdefined by f(x, y ) := (y/|y|)px2+y2if y6= 0, and f(x, y) := 0 if y= 0.
Show that fis continuous at (0,0),D~v f|(0,0) exists for every unit vector ~v in R2, but fis not
differentiable at (0,0).
7. show that the function f:R2−→ Rdefined by f(x, y) = x2y2
x4+y2for (x, y)6= (0,0) and f(0,0) = 0
is differentiable at (0,0).
8. Starting from (1,1), in which direction should one travel in order to obtain the most rapid rate of
decrease of the function f:R2−→ Rdefined by f(x, y) := (x+y−2)2+ (3x−y−6)2?
9. About how much will the function f(x, y) := lnpx2+y2change if the point (x, y)is moved from
(3,4) a distance 0.1unit straight toward (3,6)?
10. Consider f:R2−→ Rdefined by f(x, y ) := (x+y)/√2if x=y, and f(x, y) := 0 otherwise. Show
that Dxf|(0,0) =Dxf|(0,0) = 0 and D~v f|(0,0) = 1, where ~v = (1/√2,1/√2). Deduce that fis not
differentiable at (0,0).
11. Let f:R2−→ RaC1-type function. Define φ(x, y) = lim
h→0
f(hx, hy)−f(0,0)
hfor all (x, y)∈R2
satisfying x2+y2= 1. Prove that the function φexits, i.e., the given limit exists. Show that for any
constant α∈R, the level curve φ(x, y) = αrepresents a straight line. Find the normal vector at any
point of this level curve.
12. Find the directional derivative, if exists, of the given function in the given point in the indicated
direction
(i) x2y−y2z−xyz, at (1,−1,0) in the direction (ˆ
i−ˆ
j+ 2ˆ
k), (ii) (x2+y2+z2)3
2, at (−1,1,2) in
the direction (ˆ
i−2ˆ
j+ˆ
k), (iii) ex−yz, at (1,1,1) in the direction (ˆ
i−ˆ
j+ˆ
k).
13. Let h(x, y) = 2e−x2+e−3y2denote the height on a mountain at position (x, y). In what direction
from (1,0) should one begin walking in order to climb the fastest?
