Partial Differentiation?

1. Jun 23, 2012

optics.tech

Hi everyone,

I know that if

$$z = f(x,y) = x^2y + xy^2$$

then

$$\frac{\partial z}{\partial x}=2xy+y^2$$ and
$$\frac{\partial z}{\partial y}=x^2+2xy$$

Please correct me if I am wrong.

In the physics, can anyone please tell me what is the meaning of below formula?

$$\frac{\partial V}{\partial t}$$

Where V is the velocity and t is the time elapsed.

2. Jun 23, 2012

chiro

Hey optics.tech.

Both calculations are correct, and if you were thinking about keeping the other variable constant will differentiating the other variable, then your thinking is correct. The case when you can not do this is when the two variables are not independent. If they are dependent, then you can write y in terms of x (or the other way around) and you end up getting an equation in terms of 1 independent variable instead of 2. Just thought you should keep this in mind for future problems.

The rate of change of velocity with respect to time is typically known as acceleration. V can be a vector or it can be a scalar depending on the context (usually treating it as a vector is what happens unless you are learning for the first time).

It tells us how velocity changes over time instantaneously: in other words how it either increases or decreases instanteously at every particular point in time that it is defined for.

3. Jun 24, 2012

azizlwl

V=f(r,θ)

4. Jun 24, 2012

arildno

"V" is in this case typically the "velocity field", rather than the (particle) velocity.

Thus, the partial differentiation of V with respect to time does NOT equal the acceleration of the particular material particle inhabiting some fixed position at the point of time.

Rather, the partial diff of V wrt time is the locally measured rate of change of velocity for different particles inhabiting the same fixed position at different times.
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Thus, if you put a velocimeter at a fixed point in a moving stream, the rate of change of the velocity read from that apparatus equals the partial diff of V wrt. to time.
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The particle acceleration can be found from the velocity field by adding together a) this local rate of change of velocity and b) the convective acceleration term.