MAT133Y5 Chapter 10: CHAPTER 10 multivariable calculus
CHAPTER 10-MULTIVARIABLE CALCULUS
SUMMARY
In general, function snow accept multiple parameters ,such as
f(x,y,z)=xy+x2−z2. To evaluate such function, we need to be give a 3-
tuple, and substitute this into f:
f(1,2,3)=(xy+x2 −z2)x=1,y=2,z=3 =(1)(2)+(1)2 −(3)2 =2+1−9=−6.
In the special case of a two-parameter function, we can still visualize it
in terms of a three-
dimensional graph z = f (x, y)
Partial Derivatives
It’s no longer possible to describe the “slope” of a plane, so the best we
can do is talk about the slope of a line in a particular direction.
This brings about the definition of a partial derivative.
Applications of Partial Derivatives
just as in the single variable case, slapping a derivative on a quantity is
economically known as the marginal value of that quantity. For
example, if C(x,y) describes the joint cost for a manufacture to produce
two products with quantities x and y then represent the marginal cost to
produce x and the marginal cost to produce y respectively. Intuitively,
these represent the rate of change of x and y if the other variable is held
constant.
Higher-Order Partial Derivatives
For differentiable functions of one variable, a lot of information about f
could be derived not only from its first derivative f′, but from its higher
order derivatives f(n). For example, if f represents some physical
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