The function $\vc{f}(x,y) = (x+y,x-y)$ has two outputs, $x+y$ and $x-y$ in response to two inputs $x$ and $y$. For example $\vc{f}(3,1) = (3+1,3-1) = (4,2)$. Calculate the following.
$\vc{f}(0,1) = ($$+$, $-$$)$ $= ($, $)$
$\vc{f}(5,2) = $
$\vc{f}(a,b) = $
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A vector-valued function can be thought of a combination of multiple functions. We could view the above function $\vc{f}=(x+y,x-y)$ as two functions, one for each of its outputs: $f_1(x,y)=x+y$ and $f_2(x,y)=x-y$. Since each function is a component of the output vector, we call $f_1$ and $f_2$ the component functions of $\vc{f}$.
What are the component functions of $\vc{g}(x,y) = (x \ln{\left (x + y \right )},x^{2} - y)$?
$g_1(x,y) = $
$g_2(x,y) = $
What are the component functions of $\displaystyle \vc{h}(x,y) =\left ( 2 x^{2} e^{x y} - x, \quad \frac{x^{2} - y}{y - 3}\right )$? $h_1=$ $h_2=$
What is $\vc{h}(1,0)$?
Since $\vc{h}(1,0)$ is a vector, be sure to write it as an ordered pair, like $(2,3)$.
$\displaystyle\pdiff{f}{x}=$
$\displaystyle\pdiff{f}{y}=$
$\displaystyle \pdiff{f_1}{x} =$ , $\displaystyle \pdiff{f_1}{y} =$
$\displaystyle \pdiff{f_2}{x} =$ , $\displaystyle \pdiff{f_2}{y} =$
$\displaystyle \pdiff{g_1}{x} =$ , $\displaystyle \pdiff{g_1}{y} =$
$\displaystyle \pdiff{g_2}{x} =$ , $\displaystyle \pdiff{g_2}{y} =$
A matrix is array of numbers or expressions. A $2 \times 2$ matrix has two rows and two columns. We could write a $2 \times 2$ matrix as \begin{gather*} \begin{pmatrix} 1 & 2\\ 3 & 4 \end{pmatrix} \quad \text{or} \quad \begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix}. \end{gather*}
We can write the four partial derivatives of $\vc{f}(x,y)$ into a $2 \times 2$ matrix. We call this matrix either the Jacobian matrix or the matrix of partial derivatives (two names for the same matrix).
Since $\vc{f}(x,y) = (x^{2} + y,3 y^{2})$, this means $f_1(x,y)=x^{2} + y$ and $f_2(x,y)=3 y^{2}$.
When we form the matrix of partial derivatives of $\vc{f}$ (or Jacobian matrix), we write the derivatives in the same order as above. The matrix is $$D\vc{f}(x,y) = \begin{bmatrix}\textstyle\pdiff{f_1}{x} & \textstyle\pdiff{f_1}{y}\\ \textstyle\pdiff{f_2}{x} & \textstyle\pdiff{f_2}{y}\end{bmatrix}.$$ The first row contains the derivatives of the first component $f_1$; the second row contains the derivatives of $f_2$. The columns correspond to the variables, $x$ or $y$, that we differentiate with respect to.
Write the matrix of partial derivatives of $\vc{f}(x,y) = (x^2+y, 3y^2)$.
Calculate the Jacobian matrix of $\vc{g}(x,y) =(x^{2} y^{3},5 x - 3 y^{2} - 1)$.