A lot of confusion about partial derivatives

[sebastian tindall]

Homework Statement

Hi there,

what is the difference between the partial derivative and the total derivative? how do we get the gradient "the actual gradient scalar value" at a point on a multivariable function? what does the total derivative tell us and what does the partial derivative tell us.

it is my understanding that the partial derivative gives a function called the "gradient" which is a vector field of vectors that point in the direction of greatest ascent at that point in space.

does the total derivative give us a scalar field? since if it is the same as the derivative of a single variable function it should just tell us the gradient value at that point which is a scalar?

please can you explain to me why the partial derivative of a function that is written in spherical coordinates is equal to:

https://www.flickr.com/photos/148103454@N02/shares/2U26RG

i'm extremely confused

thanks

sebastian

Homework Equations

The Attempt at a Solution

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[FactChecker]

how do we get the gradient "the actual gradient scalar value" at a point on a multivariable function?

This is the first problem. The function probably has different slopes as you go in different directions of the independent variables. So there is no single number for the slope.

what does the total derivative tell us and what does the partial derivative tell us.

With multivariable functions, there is a vector of partial derivatives with the slope of the function in the direction of each independent variable. This is called the "gradient". If you pick a direction, that gives you a certain combination of changes of the independent variables. If you parameterize that direction using a new variable, t, the function will have a "total derivative" with respect to the variable t.

it is my understanding that the partial derivative gives a function called the "gradient" which is a vector field of vectors that point in the direction of greatest ascent at that point in space.

The gradient tells what the function slope is in all directions. You can use it to determine which direction of the independent variables will give you the steepest slope of the function.

 

[Grinkle]

Not a very mathematically sound attempt, hope it helps.

For a function of more than one variable, the function value is changing according to more than one variable. The partial derivative measure the change of the function with respect to one single one of those variables.

If all variables in the function also change with a common variable, say time, for instance, then one can take the total derivative of the function with respect to that common variable. I never had to solve a problem where the total derivative wasn't with respect to time, but I guess it needn't be.

So if the speed of a car changes with grade of the road as one variable and how much one is pressing the gas pedal as a second variable, one can take a partial derivative of the car speed with respect to either of those two variables, and this is taken assuming the other variable is not changing, is constant.

If both the road grade and the gas pedal press are also known functions of time, one can then get the total derivative with respect to time of the car speed.

edit:

My example is physically confusing, since for a real road the road grade depends on the cars position on the road. To just focus on the math, picture a car on a treadmill that automatically keeps exact pace with the car, and the road grade is a known function of time that is programmed into the treadmill and can change independently of the cars speed or distance traveled.

 

[sebastian tindall]

If you pick a direction, that gives you a certain combination of changes of the independent variables. If you parameterize that direction using a new variable, t, the function will have a "total derivative" with respect to the variable t.

i was following your explanation until i got to this part,

I don't know how to input the symbols correctly on this forum but the total derivative in my understanding is (for spherical coords) delf_rdr + delf_thetadtheta + delf_phidphi

so if I have a "direction" as you call this, would this just be values of r, theta and phi that are all variables of t? and what if they are simply just real numbers not functions? can I then not take the total derivative? is that when I can only take the partial derivative? does the total derivative give me a value of gradient for a movement from r1 to r2? how do these two things relate to line element integrals?

its all very jumbled up for me but thank you for your help I am trying hard to understand it :)

 

[PeroK]

i was following your explanation until i got to this part,

I don't know how to input the symbols correctly on this forum but the total derivative in my understanding is (for spherical coords) delf_rdr + delf_thetadtheta + delf_phidphi

so if I have a "direction" as you call this, would this just be values of r, theta and phi that are all variables of t? and what if they are simply just real numbers not functions? can I then not take the total derivative? is that when I can only take the partial derivative? does the total derivative give me a value of gradient for a movement from r1 to r2? how do these two things relate to line element integrals?

its all very jumbled up for me but thank you for your help I am trying hard to understand it :)

I think you have probably got all the terminology mixed up somewhere. I would start again. There's nothing wrong with this. If I find myself in a muddle, I like to go back to square one and start again. Do you have a textbook? If not, where are you getting your terminology from?

 

[FactChecker]

i was following your explanation until i got to this part,

I don't know how to input the symbols correctly on this forum but the total derivative in my understanding is (for spherical coords) delf_rdr + delf_thetadtheta + delf_phidphi

To keep things simple for now, let's work in the XY plane. Suppose we have a function f(x,y). At point (x0, y0), there is a total derivative, df/dx in the X direction (y is not changing) and there is another total derivative, df/dy, in the Y direction (x is not changing). But what about the direction at 45 degrees where both x and y are changing and the derivatives are partial? We can parameterize that direction by using a variable t to define the path ( x0 + t/√2, y0 + t/√2). That gives us a total derivative df/dt in that direction. You will need to use the partial derivatives of f wrt x and y to define df/dt.