Dy/dx of vector valued function F(x,y)

[shippo113]

F(x,y) = F1(x,y)i + F2(x,y)j, F is a vector valued function of x,y. How to find dy/dx?

The book gives the answer as

dy/dx = (F1(x,y))/(F2(x,y)). But I do not know how to get to this answer at all. Find the partial derivative is what I can do, but how to get to dy/dx when the main function is F(x,y)?

 

[hunt_mat]

I would be tempted to look at the norm of the function and then differentiate.

 

[shippo113]

excuse me but what does norm mean?

 

[hunt_mat]

Think of it as dot product.
$$\mathbf{F}(x,y)=F_{1}(x,y)\mathbf{i}+F_{2}(x,y) \mathbf{j} \Rightarrow |\mathbf{F}(x,y)|^{2}=F_{1}(x,y)^{2}+F_{2}(x,y)^{2}$$

 

[LCKurtz]

This appears to be just a vector function of two apparently independent variables x and y. Not a situation where you would normally even ask for dy/dx. What is the context of your question?

 

[shippo113]

OK so this question is from the book called "Div, Grad, Curl, and all that: an informal text on vector calculus" by h. m. Schey. This is the the question 1-6 from the book.

The question asks to prove that the F(x,y) = F1(x,y)i + F2(x,y) is infact a solution to the differential equation dy/dx=(F2(x,y))/(F1(x,y)).

Sorry but I just realized from the small print in the ebook that the dy/dx=(F2(x,y))/(F1(x,y)) and not upside down.

 

[HallsofIvy]

F(x,y) = F1(x,y)i + F2(x,y)j, F is a vector valued function of x,y. How to find dy/dx?

The book gives the answer as

dy/dx = (F1(x,y))/(F2(x,y)). But I do not know how to get to this answer at all. Find the partial derivative is what I can do, but how to get to dy/dx when the main function is F(x,y)?

The question, as given, makes no sense. In order to find dy/dx, y must be a function of x. But just telling us that "F(x,y)= F1(x,y)i+ F2(x,y)j" does not imply that. It might well be that x and y are completely independent parameters.

Please check your book again and tell us what the problem really says.

 

[LCKurtz]

It's always a challenge to figure out what the problem is when the poster doesn't understand it will enough to state it correctly. So here goes a wild guess. First I'm guessing that what the OP called F1(x,y) and F2(x,y) are really F₁(x,y) and F₂(x,y) and are alternate notations for the partials F_x(x,y) and F_y(x,y). Then I'm going to further guess that his question has something to do with the fact that if you differentiate F(x,y) = C implicitly with respect to x you get

F_x(x,y) + F_y(x,y)y' = 0

$$y' = -\frac{F_x(x,y)}{F_y(x,y)}$$