Page 376 - Advanced engineering mathematics

356 CHAPTER 11 Vector Differential Calculus

with c an arbitrary constant. Next integrate
dy
=−dz
2y
to get

1
ln|y|=−z + k.
2
It is convenient to express two of the variables in terms of the third. If we write x and y in terms
of z we have
1
x = and y = ae −2z ,
z − c
in which a is constant. This gives us parametric equations of the streamlines, with z as the
parameter. If we want the streamline through a particular point, we must choose a and c accord-
ingly. For example, suppose we want the streamline through (−1,6,2). Then z = 2 and we
need
1
−4
−1 = and 6 = ae .
2 − c
4
Then c = 3 and a = 6e so the streamline through (−1,6,2) has parametric equations
1
x = , y = 6e 4−2z , z = z.
z − 3

SECTION 11.3 PROBLEMS

In each of Problems 1 through 6, find the streamlines of 4. F = cos(y)i + sin(x)j;(π/2,0,−4)
the vector field and also the streamline through the given
z
point. 5. F = 2e i − cos(y)k;(3,π/4,0)
2
3
2
1. F = i − y j + zk;(2,1,1) 6. F = 3x i − yj + z k;(2,1,6)
2. F = i − 2j + k;(0,1,1) 7. Construct a vector field whose streamlines are circles
x
3. F = (1/x)i + e j − k;(2,0,4) about the origin.

11.4 The Gradient Field

Let ϕ(x, y, z) be a real-valued function of three variables. In the context of vector fields, ϕ
is called a scalar field.The gradient of ϕ is the vector field
∂ϕ ∂ϕ ∂ϕ
∇ϕ = i + j + k.
∂x ∂y ∂z
The symbol ∇ϕ is read "del ϕ" and ∇ is called the del operator.If ϕ is a function of just
(x, y), then ∇ϕ is a vector field in the plane.
∇ is also often called nabla.

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October 15, 2010 16:11 THM/NEIL Page-356 27410_11_ch11_p343-366