Gradient Vectors: Understanding the Operation - Homework Help

[bobsmith76]

Homework Statement

My textbook never explains well so I have to figure out how to do problems by reverse engineering using the solution manual. However, here is one operation that I simply cannot reverse engineer. I do not see a common pattern in these four problems. I can't figure out what operation is going on here. In the first one it looks like they're just multiplying i by x and j by y which would work but given the other 3 examples, that's not what's happening. I understand all the other steps but this is one operation that I don't understand.

here are the full problems in case you need to see more context.

 

[LCKurtz]

An appropriate response really depends on what level class you are in. One approach, which works on these problems is to remember if you have an equation $f(x,y)=c$, which determines y implicitly as a function of x, you have the formula$$
\frac{dy}{dx}= -\frac{f_x}{f_y}$$For example, in your problem 2 this would give$$
\frac{dy}{dx}=-\frac x y$$Have you had any differential equations so you can solve this by separation of variables? Like I said, that's not the only way, but I don't know what you have to work with.

 

[bobsmith76]

I figure out the operation. You just go back to the original question and plug in the values for x and y into the original equation, simple as that.