The document discusses gradients and divergence of scalar and vector fields. It defines the gradient of a scalar field f as the vector of its partial derivatives, representing the maximum rate of change of f in a given direction. The divergence of a vector field v is defined as the trace of its partial derivative matrix. Examples are given of computing the gradient of scalar fields like f(x,y,z)=5x^2-2xy+y^2-4yz+z^2+3xz and the divergence of vector fields like F=x^2yi-(z^3-3x)j+4y^2k. Basic properties of the gradient and divergence operators are also outlined.

1. JULIET S. ANINO
MST-MATH
ADVANCED CALCULUS

2. Let a scalar field f be given in space and a coordinate system chosen so that f = f
( x ,y , z ) and is defined and differentiable in a certain domain of space; thus the
first partial derivatives of f exist in this domain. They form the components of the
vector grad f , the gradient of the scalar f . Thus one has
For example, if f = x2y - z2, then
Formula (3.5) can be written in the following symbolic form:
where the suggested multiplication actually leads to a differentiation. The
expression
in parentheses is denoted by the symbol ∇ and is called "del" or "nabla." Thus

3. 𝛁 is a "vector differential operator." By itself, the 𝛁 has no numerical
significance; it takes on such significance when it is applied to a function, that
is, in forming
It was shown in Section 2.14 that the directional derivative of the scalar f in the
direction of the unit vector u = cos𝜶i+cos 𝜷j +cos 𝛾 k is given by
This shows that grad f has a meaning independent of the coordinate system
chosen:
It’s component in a given direction represents the rate of change off in
that direction.
In particular, grad f points in the direction of maximum increase of f.

4. The gradient obeys the following laws:
That is, with the 𝛁 symbol,
These hold, provided grad f and grad g exist in the domain considered.
If f is a constant c, (3.1 1 ) reduces to the simpler condition
If the terms in z are dropped, the preceding discussion specializes at once to two
dimensions. Thus for f = f( x , y), one has

5. The gradient obeys the following laws:
That is, with the 𝛁 symbol,
These hold, provided grad f and grad g exist in the domain considered.
If f is a constant c, (3.1 1 ) reduces to the simpler condition
If the terms in z are dropped, the preceding discussion specializes at once to two
dimensions. Thus for f = f( x , y), one has

6. Example. Find the gradient of the following scalar
fields:
1. f (x, y) = x2 y3
grad 𝑓 =
𝜕𝑓
𝜕𝑥
i +
𝜕𝑓
𝜕𝑦
j
grad f =
𝜕
𝜕𝑥
𝑥2
𝑦3
i +
𝜕
𝜕𝑦
(𝑥2
𝑦3
)j
grad f = 2xy3i + 3x2y2j
2. f(x, y, z)=5x2−2xy+y2−4yz+z2+3xz
grad 𝑓 =
𝜕𝑓
𝜕𝑥
i +
𝜕𝑓
𝜕𝑦
j +
𝜕𝑓
𝜕𝑧
k
grad f =
𝜕
𝜕𝑥
5x2−2xy+y2−4yz+z2+3xz i +
𝜕
𝜕𝑦
(5x2−2xy+y2−4yz+z2+3xz)j
+
𝜕
𝜕𝑧
(5x2−2xy+y2−4yz+z2+3xz)j
grad f = (10x-2y+3z)i + (-2x+2y-4z)j + (-4y+2z+3x)k

8. Given a vector field v in a domain D of space, one has (for a given coordinate system)
three scalar functions 𝑣𝑥, 𝑣𝑦, 𝑣𝑧. If these possess first partial derivatives in D, we can
form in all nine partial derivatives, which we arrange to form a matrix:
From three of these the scalar div v, the divergence of v, is constructed by the
formula:
It will be noted that the derivatives used form a diagonal (principal diagonal) of the
matrix and the divergence is the trace.
For example, if v = x2i – xyj + xyzk,then

9. Formula (3.15)can be written in the symbolic form:
For, treating nabla as a vector, one has
The divergence has the basic properties:
That is, with the nabla symbol:

10. 1. Compute div 𝐹 for 𝐹 = x2yi – (z3-3x)j + 4y2k
 2.
= 3 +
2𝑥3𝑦
𝑧
- 1 = 2 +
2𝑥2𝑦2
𝑧