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------------------------------------------------ The domain of [tex]f(x, y)[/tex] is the [tex]xy[/tex]-plane, and values of [tex]f[/tex] are given in the table below.

\[
\begin{array}{|c|c|c|c|c|c|}
\hline
y \backslash x & 0 & 1 & 2 & 3 & 4 \\
\hline
0 & 65 & 64 & 63 & 64 & 65 \\
\hline
1 & 64 & 63 & 62 & 63 & 62 \\
\hline
2 & 60 & 59 & 60 & 61 & 60 \\
\hline
3 & 59 & 59 & 59 & 61 & 61 \\
\hline
4 & 60 & 59 & 60 & 61 & 62 \\
\hline
\end{array}
\]

Find [tex]\int_C \operatorname{grad} f \cdot d \vec{r}[/tex], where [tex]C[/tex] is:

(a) A line from [tex](0,2)[/tex] to [tex](2,3)[/tex].

\[ \int_C \operatorname{grad} f \cdot d \vec{r} = \square \]

(b) A circle of radius 1 centered at [tex](3,1)[/tex] traversed counterclockwise.

\[ \int_C \operatorname{grad} f \cdot d \vec{r} = \square \]

Answer :

To solve this problem, we want to find the line integrals of the gradient of function [tex]\( f(x, y) \)[/tex] over two different paths [tex]\( C \)[/tex].

### Part (a)

Path: A straight line from [tex]\( (0, 2) \)[/tex] to [tex]\( (2, 3) \)[/tex].

1. Understand the Problem: We're given values of the function [tex]\( f(x, y) \)[/tex] in a table at discrete points. We need to calculate the line integral of the gradient of [tex]\( f \)[/tex] along a straight path [tex]\( C \)[/tex].

2. Use the Fundamental Theorem of Line Integrals: The theorem states that for a conservative vector field, the line integral of a gradient over a path depends only on the values of the scalar potential function [tex]\( f \)[/tex] at the endpoints of the path:
[tex]\[
\int_C \operatorname{grad} f \cdot d \vec{r} = f(x_2, y_2) - f(x_1, y_1)
\][/tex]

3. Apply to the Points:
- Start Point: [tex]\((0, 2)\)[/tex]
- End Point: [tex]\((2, 3)\)[/tex]

4. Look Up Values in the Table:
- [tex]\( f(0, 2) = 60 \)[/tex]
- [tex]\( f(2, 3) = 59 \)[/tex]

5. Compute the Line Integral:
[tex]\[
\int_C \operatorname{grad} f \cdot d \vec{r} = f(2, 3) - f(0, 2) = 59 - 60 = -1
\][/tex]

### Part (b)

Path: A circle of radius 1 centered at [tex]\( (3, 1) \)[/tex] traversed counterclockwise.

1. Understand the Path: The path [tex]\( C \)[/tex] is a closed loop, specifically a circle.

2. Key Property of Conservative Fields: For a closed loop in a conservative vector field, the line integral is zero because the net change in the potential over a closed path is zero.

3. Conclusion for the Integral Over the Circle:
[tex]\[
\int_C \operatorname{grad} f \cdot d \vec{r} = 0
\][/tex]

### Summary

- For path (a), the line integral is [tex]\(-1\)[/tex].
- For path (b), the line integral is [tex]\(0\)[/tex].

These results are based on the properties of line integrals and the specific values given in the table for the function [tex]\( f(x, y) \)[/tex].