Motivation to Create the Total Derivative

[V0ODO0CH1LD]

If I have w(x, y, z) and take "dw = (∂w/∂x)dx + (∂w/∂y)dy + (∂w/∂z)dz"; doesn't that accurately describe how w changes as x, y and z change by an infinitesimal? Or does that only work for some special cases?

I feel like if I take the total derivative I am actually describing how the function w changes with respect to an infinitesimal change in a parameter t. Or to one of the variables which all the other variables happen to depend on, not the variables x, y and z. Is that right?

Also, does "dw = (∂w/∂x)dx + (∂w/∂y)dy" mean that if I go one unit in the direction of x and one unit in the direction of y I will have climbed "(∂w/∂x) + (∂w/∂y)" along the direction of w?
Which is kind of like going one for x and one for y along the tangent plane?

 

[Stephen Tashi]

If I have w(x, y, z) and take "dw = (∂w/∂x)dx + (∂w/∂y)dy + (∂w/∂z)dz"; doesn't that accurately describe how w changes as x, y and z change by an infinitesimal? Or does that only work for some special cases?

You aren't asking precise mathematical questions; you're asking for an intutitive descriptions. It's OK to ask for intuitive descriptions, but realize that the type of answer you get isn't useable in a formal mathematical proof.

To the above question, I'd say your interpretation works for functions that are "smooth". I think you could make an example of a function w(x,y,z) that has a slope along a particular line of approach to the point (x,y,z) that is different from the slope predicted by the tangent plane to the function.

I feel like if I take the total derivative I am actually describing how the function w changes with respect to an infinitesimal change in a parameter t. Or to one of the variables which all the other variables happen to depend on, not the variables x, y and z. Is that right?

I'd say "No". If you introduce a variable t and visualize x,y,z as functions of t then to find how w changes with t, you'd have to use the chain rule. So to find $\frac{dW}{dt}$ you would have a sum with terms like $\frac{\partial w}{\partial x} \frac{\partial x}{\partial t}$

Also, does "dw = (∂w/∂x)dx + (∂w/∂y)dy" mean that if I go one unit in the direction of x and one unit in the direction of y I will have climbed "(∂w/∂x) + (∂w/∂y)" along the direction of w?
Which is kind of like going one for x and one for y along the tangent plane?

Are you visualizing a Cartesian coordinate system where w has its own axis? The change in the w coordinate that you describe is only correct if it all the climbing takes place in the tangent plane. The tangent place is only an approximation for what happens if you move along the surface w.