Math Insight

1. 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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   Hint

   For $\vc{f}(5,2)$ and $\vc{f}(a,b)$, write your answer as an ordered pair, such as $(1,2)$.

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2. 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) = $
   

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   Hint

   Online, if you need to enter something like $x \ln(\ldots)$, be sure to put a space or a * between the x and the ln, or the computer will think you meant $x \times l \times n$.

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3. 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 $\vc{h}(1,0)$?
   

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   Hint

   Online, when writing the fraction for $h_2$, don't forget to put parenthesis around the numerator and around the denominator.

   Since $\vc{h}(1,0)$ is a vector, be sure to write it as an ordered pair, like $(2,3)$.

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1. Your typical function $f(x)=x^2$ has one derivative; it is $\diff{f}{x}=2x$. How many partial derivatives does the multivariable function $f(x,y)=x^{2} - y^{2}$ have?
   What are they?

   $\displaystyle\pdiff{f}{x}=$
   

   $\displaystyle\pdiff{f}{y}=$
   

2. A vector-valued function of the form $\vc{f}(x,y) = \left ( x^{2} y, \quad x y^{2}\right )$ has how many component functions?
   What are they?
   $f_1(x,y)=$
   
   $f_2(x,y)=$
   
3. Each of the two component functions of $\vc{f}(x,y) =\left ( x^{2} y, \quad x y^{2}\right )$ has how many partial derivatives?
   This means that the vector-valued function $\vc{f}$ has
   $\times$
   =
   partial derivatives.
4. What are the partial derivatives of the function $\vc{f}(x,y) = \left ( x^{2} y, \quad x y^{2}\right )$?

   $\displaystyle \pdiff{f_1}{x} =$
   , $\displaystyle \pdiff{f_1}{y} =$
   

   $\displaystyle \pdiff{f_2}{x} =$
   , $\displaystyle \pdiff{f_2}{y} =$
   

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   Hint

   Take the partial derivatives of the functions $f_1$ and $f_2$ that you found in part b.

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5. What are the partial derivatives of $\vc{g}(x,y) =\left ( x^{2} + y^{2}, \quad x - y\right )$?

   $\displaystyle \pdiff{g_1}{x} =$
   , $\displaystyle \pdiff{g_1}{y} =$
   

   $\displaystyle \pdiff{g_2}{x} =$
   , $\displaystyle \pdiff{g_2}{y} =$
   

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   Hint

   The function $g_1(x,y)$ is the first component of $\vc{g}(x,y)$; the function $g_2(x,y)$ is the second component of $\vc{g}(x,y)$;

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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).

1. Let $\vc{f}(x,y) = (x^{2} + y,3 y^{2})$. What are the four partial derivatives of $\vc{f}$?

   $\displaystyle \pdiff{f_1}{x} =$
   , $\displaystyle \pdiff{f_1}{y} =$
   

   $\displaystyle \pdiff{f_2}{x} =$
   , $\displaystyle \pdiff{f_2}{y} =$
   

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   Hint

   Note: one of the partial derivatives is zero.

   Since $\vc{f}(x,y) = (x^{2} + y,3 y^{2})$, this means $f_1(x,y)=x^{2} + y$ and $f_2(x,y)=3 y^{2}$.

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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 $\vc{f}(x,y) = (x^2+y, 3y^2)$.

   
   
   $D\vc{f}(x,y)=$
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   Hint

   You should write the same answers as in the previous part, just organized into a matrix.

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3. Calculate the Jacobian matrix of $\vc{g}(x,y) =(x^{2} y^{3},5 x - 3 y^{2} - 1)$.

   
   
   $D\vc{g}(x,y)=$
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   Hint

   Follow the same procedure as for $\vc{f}$. In the first row, enter the partial derivatives of $g_1(x,y)=x^{2} y^{3}$. In the second row, enter the partial derivatives of $g_2(x,y)=5 x - 3 y^{2} - 1$. The first column contains partial derivatives with respect to $x$; the second column contains partial derivatives with respect to $y$.

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1. $\vc{f}(x,y) = \left ( e^{x} + e^{y}, \quad e^{x + y}\right )$
   
   
   $D\vc{f}(x,y)=$
2. $\vc{g}(x,y) = \left ( x \ln{\left (y \right )}, \quad y \ln{\left (y \right )}\right )$
   
   
   $D\vc{g}(x,y)=$
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   Hint

   The four partial derivatives may look similar, but they are all quite different. Some are quite simple to calculate, especially if the variable inside the logarithm is viewed as a constant. Others might require the product rule.

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3. $\vc{h}(x,y) = \left ( \ln{\left (x y \right )}, \quad x e^{x y}\right )$
   
   
   $D\vc{h}(x,y)=$
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   Hint

   The product rule and/or the chain rule will be helpful here.

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