Gradient: denition and properties
Denition of the gradient
w
w
If w = f (x, y), then
and
are the rates of change of w in the i and j directions.
x
y
It will be quite useful to put these two derivatives together in a vector called the gradient
of w.
w w
,
The Tangent Approximation
1. The tangent plane.
w
For a function of one variable, w = f (x), the tangent line to its graph(
)
dw
.
at a point (x0 , w0 ) is the line passing through (x0 , w0 ) and having slope
dx 0
w=f(x,y)
w=f(x,y 0)
For a function of two
Chain rule with more variables
1. Let w = xyz, x = u2 v, y = uv 2 , z = u2 + v 2 . w a) Use the chain rule to find . u b) Find the total differential dw in terms of du and dv. w c) Find at the point (u, v) = (1, 2). u Answer: a) The chain rule says w u =