partial derivatives

Qing, thanks for providing nice questions all the time.

圖片參考：https://s.yimg.com/rk/HA00430218/o/1732979963.gif

(a)

1. For fy(B), suppose x is fixed at 2, look at the curved path from back to front along with the direction from -ve y to +ve y, we can see that it is "downhill" at B, so the answer is fy(B) < 0.

2. For fx(A), suppose y is fixed at 2, look at the curved path from right to left along with the direction from -ve x to +ve x, we can see that it is "uphill" at A, so the answer is fx(A) > 0.

3. For fy(A), suppose x is fixed at -2, look at the curved path from back to front along with the direction from -ve y to +ve y, we can see that it is "downhill" at A, so the answer is fy(A) < 0.

4. For fx(B), suppose y is fixed at 2, look at the curved path from right to left along with the direction from -ve x to +ve x, we can see that it is "downhill" at B, so the answer is fx(B) < 0.


(b)

1. Look at (a) 3. and (a) 1., using the same idea, look at the curved path from back to front along with the direction from -ve y to +ve y. Every such path is "downhill" at the point along a path from A to B. Note that this "downhill" is looked from the path from back to front, while "the path from A to B" is the inverted U shaped path first going up then going down.
Since it is always "downhill", the sign of fy(P) from A to B is always negative, no change in sign.

2. Look at (a) 2. and (a) 4., using the same idea, look at the curved path from right to left along with the direction from -ve x to +ve x. Such path is first "uphill" and then "downhill". Therefore, the sign of fx(P) from A to B starts from a positive value at A, then reaches zero (when the point arrives the peak), and finally ends at a negative value at B. As a result, the sign of fx(P) changes from positive to negative.

For 1. and 2. in (b), you can also use the results in (a) to help you think.

2014-09-25 18:34:45 補充：
［字數滿額，請看意見欄］

2014-09-25 18:34:52 補充：
補充回答：

你問得好好，我們一定要從一個變數的負開始移向正。
因為我們考慮的是當一個變數 increase 的時候怎樣怎樣。
因此，方向必定是由小到大，即由負到正。

而圖中
back = -ve y
front = +ve y
所以看 y 的 increment 必定是 from back to front

另外，
right = -ve x
left = +ve x
所以看 x 的 increment 必定是 from right to left

2014-09-25 18:35:58 補充：
但小心圖中的 axis 的方向不是獨特的，即是說，下次你見到另一幅圖，可能是 from front to back, from left to right，要視乎那個 axis 的變數由小至大的方向。