1. f(x, y) = 1
3
x
3 − 3x
2 +
y
2
4 + xy + 13x − y + 2

2. f(x, y, z) = x
3 − 2x
2 + y
2 + z
2 − 2xy + xz − yz + 3z

3. f(x, y) = x
3 + y
5 − 3x − 10y + 4

4. f(x, y) = 1
3
x
3 − x −
1
3
y
3 + y

5. f(x, y) = 2x
3 + 4y
2 − 2y
4 − 6x

6. f(x, y, z) = x
2
y − yez + 2x + z

7. f(x, y) = −
1
2
xy +
2
x −
1
y

8. f(x, y) = (x
2 − y
2
)e
−(x
2+y
2
)/2

9. f(x, y) = e
−(x
2+y
2
)
(x
2 + 2y
2
)

You seem to be listing several functions of two or three variables, possibly for analysis or differentiation purposes. These functions involve polynomials, exponentials, and other expressions. Let’s take a closer look at each one:

1. **\( f(x, y) = \frac{1}{3}x^3 - 3x^2 + \frac{y^2}{4} + xy + 13x - y + 2 \)**  
   This function contains cubic, quadratic, and mixed terms in \(x\) and \(y\), along with constants.

2. **\( f(x, y, z) = x^3 - 2x^2 + y^2 + z^2 - 2xy + xz - yz + 3z \)**  
   A function involving three variables (\(x, y, z\)) with cubic and quadratic terms, as well as mixed terms like \(xy\), \(xz\), and \(yz\).

3. **\( f(x, y) = x^3 + y^5 - 3x - 10y + 4 \)**  
   A function with higher-degree terms like \(x^3\) and \(y^5\), linear terms, and a constant.

4. **\( f(x, y) = \frac{1}{3}x^3 - x - \frac{1}{3}y^3 + y \)**  
   A simpler function involving cubic and linear terms in both \(x\) and \(y\).

5. **\( f(x, y) = 2x^3 + 4y^2 - 2y^4 - 6x \)**  
   A function that includes cubic, quadratic, quartic terms in \(y\), and linear terms in \(x\).

6. **\( f(x, y, z) = x^2 y - ye^z + 2x + z \)**  
   This function contains a product of \(x^2 y\), an exponential term \(e^z\), and linear terms in \(x\) and \(z\).

7. **\( f(x, y) = -\frac{1}{2}xy + \frac{2}{x} - \frac{1}{y} \)**  
   A rational function involving terms like \( \frac{2}{x} \) and \( \frac{1}{y} \), plus a mixed term in \(xy\).

8. **\( f(x, y) = (x^2 - y^2)e^{-(x^2 + y^2)/2} \)**  
   This function includes a quadratic term multiplied by an exponential decay factor involving both \(x\) and \(y\).

9. **\( f(x, y) = e^{-(x^2 + y^2)}(x^2 + 2y^2) \)**  
   Similar to the previous function but with different factors in the polynomial and the exponential.

### Would you like help with:
1. Finding the partial derivatives of these functions?
2. Analyzing the critical points or extrema?
3. Evaluating these functions for specific values of \(x\), \(y\), and \(z\)?
4. Finding gradient and Hessian matrices?
5. Plotting these functions for better visualization?

**Tip**: When working with multivariable functions, partial derivatives give insights into how the function behaves as each variable changes independently.

1. f(x, y) = (1/3)x^3 − 3x^2 + (y^2)/4 + xy + 13x − y + 2
2. f(x, y, z) = x^3 − 2x^2 + y^2 + z^2 − 2xy + xz − yz + 3z
3. f(x, y) = x^3 + y^5 − 3x − 10y + 4
4. f(x, y) = (1/3)x^3 − x − (1/3)y^3 + y
5. f(x, y) = 2x^3 + 4y^2 − 2y^4 − 6x
6. f(x, y, z) = x^2y − yez + 2x + z
7. f(x, y) = −(1/2)xy + (2/x) − (1/y)
8. f(x, y) = (x^2 − y^2)e^−((x^2 + y^2)/2)
9. f(x, y) = e^−(x^2 + y^2)(x^2 + 2y^2)

Explore a detailed analysis of multivariable functions, including cubic, quadratic, and exponential terms. Learn about partial derivatives, gradients, and the Hessian matrix. Suitable for advanced calculus students.

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