Partial Derivative of Vectors a and b with Respect to x

Thread starter math_trouble
Start date May 6, 2011
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Derivative Partial Partial derivative

In summary, a simple partial derivative is a mathematical concept used to measure the instantaneous rate of change of a function with respect to one of its variables. It is calculated by taking the limit of the change in the function divided by the change in the variable as the change in the variable approaches zero. This concept is different from a total derivative, which takes into account changes in all variables simultaneously. Simple partial derivatives have many real-world applications in fields such as physics, chemistry, economics, and engineering. However, they can be undefined if the function is not continuous or differentiable at a certain point.

May 6, 2011

math_trouble

Lets say I am having 2 vector a(x,y) and b(x,y)
and i were to take :
1)the partial derivative of a(x,y) with respect x multiply by b(x,y)
- b*(da/dx)
will this be equals to a*(db/dx)

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May 6, 2011

tiny-tim

Science Advisor

Homework Helper

hi math_trouble!

(have a curly d: ∂

)

are you asking whether b∂a/∂x = a∂b/∂x ?

suppose a(x,y) = x and b(x,y) = y, then that's y*1 = x*0 :redface:

(why did you think it would work?

)

Related to Partial Derivative of Vectors a and b with Respect to x

1. What is a simple partial derivative?

A simple partial derivative is a mathematical concept used to measure the instantaneous rate of change of a function with respect to one of its variables. It is similar to a regular derivative, but only considers changes in one variable while holding all other variables constant.

2. How is a simple partial derivative calculated?

A simple partial derivative is calculated by taking the limit of the change in the function divided by the change in the variable as the change in the variable approaches zero. This can be represented mathematically as ∂f/∂x = lim(h→0) (f(x+h) - f(x)) / h, where ∂f/∂x represents the simple partial derivative of f with respect to x.

3. What is the difference between a simple partial derivative and a total derivative?

A simple partial derivative measures the rate of change of a function with respect to one variable, while holding all other variables constant. In contrast, a total derivative takes into account changes in all variables simultaneously. This means that a total derivative is more comprehensive, but also more complex to calculate.

4. What are some real-world applications of simple partial derivatives?

Simple partial derivatives are used in many fields of science and engineering, such as physics, chemistry, economics, and engineering. They can help us understand the rate of change of physical quantities, such as velocity and acceleration, in relation to other variables. They are also used in optimization problems, where we want to find the maximum or minimum value of a function.

5. Can a simple partial derivative ever be undefined?

Yes, a simple partial derivative can be undefined if the function is not continuous or differentiable at a certain point. This can happen if the function has a sharp corner or a discontinuity at that point. In these cases, the simple partial derivative does not exist because the limit does not exist.