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Homework answers / question archive / In single variable calculus we studied scalar-valued functions defined from R R and parametric curves in the case of R -0 R2 and R -0 R3

In single variable calculus we studied scalar-valued functions defined from R R and parametric curves in the case of R -0 R2 and R -0 R3. In the study of multivariate calculus we've begun to consider scalar-valued functions of two variables in the case R2 -0 R. Let us now try to think of all the possible functions we may come across in the study of real variable calculus.

Different Types of Functions:

Scalar-valued functions from R to R: For example consider f : Rem R defined by f (x) = + 2x2

Scalar-valued functions of multiple variables from R. to R: For example consider f : R2 .? R defined by f (x, y) = x2 - XY2 •

Parametric curves from R to II m:

• Planar curves For example consider r : R R2 defined by

r(t) = (cos (t),sin(t)).

• Space curves For example consider r : Ilt m R3. defined by r(t) = ( cos (t), sin (t), 1÷)). Vector-valued functions of multiple variables from R. to Er: For example consider f R3 .? R2 defined by

f (x, y, z) = (3yz, 4x + y).

With all these new functions, we return to a familiar question: How do we differentiate these things? We have considered derivatives for differentiable functions from R -0 R, R R", and R"' . Recognize that these are specific cases of functions from R. -0 R..

Derivatives of Diffent Types of Functions:

Scalar-valued functions from R to R: We define the derivative of a differentiable scalar function f as

Parametric curves from R to R.: We define the derivative of any vector-valued function of one variable f R'°, for f (t) = (xi (t), , xn(t))

aS

given each x'i(t) exists.

Scalar-valued functions of multiple variables from R. to R: We define the derivative of a scalar-valued function f : R. .• R, given each partial of f exists and is continuous as

We can also refer to this derivative as the gradient vector of f , and denote the gradient of V f.

Vector-valued functions of multiple variables from R. to R"L: We define the derivative of a vector-valued function of two variables f :R3 R2, for f(x,y,x) = (u(x,y,x),v(x,y,.0) as the 2 x 3 mat r i >: Df = [ux(x, z) uy(x,Y, z) uz(x, z)1 _ iVul Lv.(x,Yt.) vy(stY,z) vz(x,Y,z) J LVvi given each partial derivative exists and is continuous.

Compute the derivative for the following functions.

1. f(x,y) = (.2 + Y2 , x )

2. g (u, v) = 2u2 - v2

3. r(t) = (3t2,1n1t1,1,cot(t))

4. w (r , s, t) = (r2 s, + 3.52)