- #1
calculusisrad
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Homework Statement
Find a function f(x,y,z) such that F = (gradient of F).
The Attempt at a Solution
I don't know :(
I'm so confused
Please help me!
Calculating Gradients with Vector Calculus
- Thread starter calculusisrad
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In summary, the conversation is about finding a function f(x,y,z) such that F=(gradient of f). The participants are initially confused about the concept of a gradient, but eventually determine that the reverse of differentiation is necessary to find f. One participant provides a possible solution in the form of F = (2xye^z)i + ((e^z)(x^2))j + ((x^2)y(e^z)+(z^2))k, and the other suggests using the definition of a gradient to find f. The conversation ends with one participant asking for assistance due to being behind on homework.
Find a function f(x,y,z) such that F = (gradient of F).
I don't know :(
I'm so confused
Please help me!
- #2
tiny-tim
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hi calculusisrad! 
do you mean "Find a function f(x,y,z) such that F = (gradient of f)" ?
(only scalars have gradients, there's no gradient of a vector)
calculusisrad said:
do you mean "Find a function f(x,y,z) such that F = (gradient of f)" ?
(only scalars have gradients, there's no gradient of a vector)
i don't understand either
is either f or F given in the question?
is either f or F given in the question?
- #3
calculusisrad
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Sorry, yes you're right. The gradient of f should not be bolded.
- #4
1MileCrash
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Think about what a gradient is. If I told you to find the gradient of a function, what would you do?
You would differentiate the function wrt x, and that is the i component of the gradient, you would differentiate the function wrt y, and that is the j component, and then you would differentiate the function wrt z, and that is the k component.
Now, we are going in reverse. What is the reverse of differentiation?
You would differentiate the function wrt x, and that is the i component of the gradient, you would differentiate the function wrt y, and that is the j component, and then you would differentiate the function wrt z, and that is the k component.
Now, we are going in reverse. What is the reverse of differentiation?
- #5
calculusisrad
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I completely forgot the biggest part of the problem. WOW. Sorry about that!
Let F = (2xye^z)i + ((e^z)(x^2))j + ((x^2)y(e^z)+(z^2))k
NOW find a function f(x,y,z) such that F = Gradient of f.
Sorry about that. Please answer :)
Let F = (2xye^z)i + ((e^z)(x^2))j + ((x^2)y(e^z)+(z^2))k
NOW find a function f(x,y,z) such that F = Gradient of f.
Sorry about that. Please answer :)
- #6
DivisionByZro
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"Please answer"? How about you show some effort first? You should have read the forums rules by now.
- #7
calculusisrad
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This was due last Thursday, I'm horribly behind on homework, I'm desperate here.
- #8
Dick
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calculusisrad said:
It's pretty easy to guess a form for f that works. Start guessing. That's often the easiest way to solve problems like this. What's a likely form for f given the first component of F?
Last edited:
- #9
HallsofIvy
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You know what the definition of "gradient" is, so use that.
[itex]\frac{\partial f}{\partial x}=[/itex] what?
[itex]\frac{\partial f}{\partial y}=[/itex] what?
[itex]\frac{\partial f}{\partial z}=[/itex] what?
[itex]\frac{\partial f}{\partial x}=[/itex] what?
[itex]\frac{\partial f}{\partial y}=[/itex] what?
[itex]\frac{\partial f}{\partial z}=[/itex] what?
FAQ: Calculating Gradients with Vector Calculus
What is a vector calculus gradient?
A vector calculus gradient is a mathematical concept used to describe the rate of change or slope of a function in multiple dimensions. It is represented by a vector that points in the direction of the steepest increase of the function.
How is the gradient calculated?
The gradient is calculated by taking the partial derivatives of a multivariable function with respect to each variable and combining them into a vector. This vector represents the direction and magnitude of the function's steepest increase.
What is the relationship between gradients and level curves?
Gradients and level curves are closely related. The gradient vector is always perpendicular to the level curve at any given point. This means that the gradient points in the direction of the steepest increase while the level curve represents points with equal values of the function.
Can gradients be used to find maximum and minimum values of a function?
Yes, gradients can be used to find maximum and minimum values of a function. The direction of the gradient vector points towards the steepest increase of the function, while the magnitude of the gradient represents the rate of change. Therefore, the maximum and minimum values of a function can be found where the gradient is zero or undefined.
How are gradients used in real-world applications?
Gradients are used in various real-world applications, such as physics, engineering, and economics. They are used to model changes in physical quantities, optimize systems, and solve problems involving multiple variables and dimensions. Some examples include predicting weather patterns, designing efficient transportation routes, and optimizing financial portfolios.