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check the picture below.
a positive slope, means an increasing slope, namely, a line that is higher from left-to-right, so the right tip of it is going upwards. A negative y-intercept,
now, if you notice the 3 green lines, the line is increasing, the right-side is higher than the left-side, and the y-intercept, namely where x = 0, or where the graph touches the y-axis, their y-intercept is on a negative value of y.
now, look at the red line, it has also a positive slope, is increasing, however, there's no way it can have negative y-intercept. If it remains increasing, it can only have a y-intercept of 3 or higher, but not lower than that. The only way it can have a negative value for the y-intercept is if it tips over, but if we do that, the slope will be decreasing.
The gradient theorem applies here, because we can find a scalar function <em>f</em> for which ∇ <em>f</em> (or the gradient of <em>f</em> ) is equal to the underlying vector field:
We have
where <em>C</em> is an arbitrary constant.
So we found
and by the gradient theorem,