Page 381 - Advanced engineering mathematics

11.4 The Gradient Field 361

5

4

–4
–2
–2
–4
–2 2 0
4 2 4

2
2
FIGURE 11.9 Circular cone z = x + y .

EXAMPLE 11.8

2
2
The level surface ϕ(x, y, z) = z − x + y is a cone with vertex at the origin (Figure 11.9).
Compute
√ 1 1
∇ϕ(1,1, 2) =−√ i − √ j + k.
2 2
√
The tangent plane to the cone at (1,1, 2) has the equations
1 1 √
−√ (x − 1) − √ (y − 1) + z − 2 = 0
2 2
or
√
x + y − 2z = 0.
√
The normal line to the cone at (1,1, 2) has parametric equations
1 1 √
x = 1 − √ t, y = 1 − √ t, z = 2 + t.
2 2

SECTION 11.4 PROBLEMS

2
2
2
In each of Problems 1 through 6, compute the gradient of 6. ϕ(x, y, z) = x + y + z ;(2,2,2)
the function and evaluate this gradient at the given point.
Determine at this point the maximum and minimum rate of In each of Problems 7 through 10, compute the direc-
change of the function at this point. tional derivative of the function in the direction of the given
vector.
1. ϕ(x, y, z) = xyz;(1,1,1)
√
2
2
2. ϕ(x, y, z) = x y − sin(xz);(1,−1,π/4) 7. ϕ(x, y, z) = 8xy − xz;(1/ 3)(i + j + k)
z
z
3. ϕ(x, y, z) = 2xy + xe ;(−2,1,6) 8. ϕ(x, y, z) = cos(x − y) + e ;i − j + 2k
3
2
4. ϕ(x, y, z) = cos(xyz);(−1,1,π/2) 9. ϕ(x, y, z) = x yz ;2j + k
5. ϕ(x, y, z) = cosh(2xy) − sinh(z);(0,1,1) 10. ϕ(x, y, z) = yz + xz + xy;i − 4k
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