Total Derivative of a Function of Two Variables

[amaresh92]

greetings,

consider a function f(x,y);
the total derivative of a function of two varible is given by-:

df=(dou)f/(dou)x*dx+(dou)f/(dou)y*dy

here we have the differential of f(x,y).but i am not able to understand why the term dx and dy has appeared?

advanced thanks.

 

[JJacquelin]

consider a function f(x,y);
the total derivative of a function of two varible is given by-:
df=(dou)f/(dou)x*dx+(dou)f/(dou)y*dy
here we have the differential of f(x,y).but i am not able to understand why the term dx and dy has appeared?

Now, consider a function f(x);
the total derivative of a function of one varible is given by-:
df=(df/dx)*dx
here we have the differential of f(x). Are you able to understand why the term dx appeared?

 

[dextercioby]

greetings,

consider a function f(x,y);
the total derivative of a function of two varible is given by-:

df=(dou)f/(dou)x*dx+(dou)f/(dou)y*dy
here we have the differential of f(x,y).but i am not able to understand why the term dx and dy has appeared?

advanced thanks.

Well, the bolded part is inaccurate, df is the differential of the function f, not the total derivative.

From a geometric perspective, if f:M -> R is a 0-form on a smooth 2-dimensional differentiable manifold M, then the 1-form df is called the exterior differential of the 0-form f and, assuming the cotangent space in the point p=p(x,y) of the manifold is spanned by the 1-forms dx and dy, the expression

$$df\left|\right_{p=p\left(x,y\right)} := \frac{\partial f}{\partial x}\left|\right_{p=p\left(x,y\right)} dx + \frac{\partial f}{\partial y}\left|\right_{p=p\left(x,y\right)} dy$$

is just a definition of the components of the 1-form in the chosen basis at the point p of M.

 

[amaresh92]

Now, consider a function f(x);
the total derivative of a function of one varible is given by-:
df=(df/dx)*dx
here we have the differential of f(x). Are you able to understand why the term dx appeared?

then why we are adding them?