Page 382 - Advanced engineering mathematics

362 CHAPTER 11 Vector Differential Calculus

2
2
2
In each of Problems 11 through 16, find the equation of 14. x − y + z = 0;(1,1,0)
the tangent plane and normal line to the level surface at the
15. 2x − cos(xyz) = 3;(1,π,1)
point.
4
4
4
16. 3x + 3y + 6z = 12;(1,1,1)
√
2
2
2
11. x + y + z = 4;(1,1, 2)
17. Suppose that ∇ϕ(x, y, z) = i + k. What can be said
2
12. z = x + y;(−1,1,2) about level surfaces of ϕ? Show that the streamlines
2
2
2
13. z = x − y ;(1,1,0) of ∇ϕ are orthogonal to the level surfaces of ϕ.
11.5 Divergence and Curl
The gradient operator produces a vector field from a scalar function. We will discuss two other
important vector operations. One produces a scalar field from a vector field, and the other
produces a vector field from a vector field. Let
F(x, y, z) = f (x, y, z)i + g(x, y, z)j + h(x, y, z)k.
The divergence of F is the scalar field
∂ f ∂g ∂h
div F = + + .
∂x ∂y ∂z
The curl of F is the vector field
∂h ∂g ∂ f ∂h ∂g ∂ f

curl F = − i + − j + − k.
∂y ∂z ∂z ∂x ∂x ∂y

Divergence, curl and gradient can all be written as vector operations with the del operator
∇, which is a symbolic vector defined by
∂ ∂ ∂
∇= i + j + k.
∂x ∂y ∂z
The symbol ∇, which is called "del", or sometimes "nabla", is treated like a vector in carrying out
calculations, and the "product" of ∂/∂x, ∂/∂y and ∂/∂z with a scalar function ϕ is interpreted to
mean, respectively, ∂ϕ/∂x, ∂ϕ/∂y and ∂ϕ/∂z. Now observe how gradient, divergence, and curl
are obtained using this operator.

1. The product of the vector ∇ and the scalar function ϕ is the gradient of ϕ:

∂ ∂ ∂
∇ϕ = i + j + k ϕ
∂x ∂y ∂z
∂ϕ ∂ϕ ∂ϕ
= i + j + k = gradient of ϕ.
∂x ∂y ∂z
2. The dot product of ∇ and F is the divergence of F:

∂ ∂ ∂
∇· F = i + j + k · ( f i + gj + hk)
∂x ∂y ∂z
∂ f ∂g ∂h
= + + = divergence of F.
∂x ∂y ∂z

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October 15, 2010 16:11 THM/NEIL Page-362 27410_11_ch11_p343-366