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I tried to find f by integrating the partial derivatives, but since 1/r is multiplying the whole vector, I just took it out, I'm not sure if I can do that. Like this:

∂f∂x(x,y,z)=x

∂f∂y(x,y,z)=y

∂f∂z(x,y,z)=z

thus

f(x,y,z)=x22+g(y,z)

f(x,y,z)=y22+h(x,z)

f(x,y,z)=z22+k(x,y)

for some functions g, h, and k, so if g=y22+z22, h=x22+z22 and k=x22+y22, the function f is:

f(x,y,z)=1r(x22+y22+z22)=12r⋅r2=r2

Am I correct? If not, how can I solve this correctly, should I integrate x/r, y/r and z/r instead?

What else can I help you with?

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To find the gradient of a function, you calculate the partial derivatives of the function with respect to each variable. For a function ( f(x, y) ), the gradient is represented as a vector ( \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) ). This vector points in the direction of the steepest ascent of the function and its magnitude indicates the rate of increase. You can compute the gradient using calculus techniques, such as differentiation.


What is a gradient function?

Assume you want to know what is the formula of the gradient of the function in multivariable calculus. Let F be a scalar field function in n-dimension. Then, the gradient of a function is: ∇F = <fx1 , fx2, ... , fxn> In the 3-dimensional Cartesian space: ∇F = <fx, fy, fz>


What is the formula of gradient?

Assume you want to know what is the formula of the gradient of the function in multivariable calculus. Let F be a scalar field function in n-dimension. Then, the gradient of a function is: ∇F = <fx1 , fx2, ... , fxn> In the 3-dimensional Cartesian space: ∇F = <fx, fy, fz>


How do you find a gradient in a equation?

To find the gradient of an equation, you typically take the derivative of the function with respect to its variable. For a function ( y = f(x) ), the gradient (or slope) at any point is given by ( f'(x) ). If the equation is in the form ( Ax + By + C = 0 ), you can rearrange it to the slope-intercept form ( y = mx + b ), where ( m ) represents the gradient. This value indicates how steeply the line rises or falls as you move along the x-axis.


What is the formula for functions?

Assume you want to know what is the formula of the gradient of the function in multivariable calculus. Let F be a scalar field function in n-dimension. Then, the gradient of a function is: ∇F = <fx1 , fx2, ... , fxn> In the 3-dimensional Cartesian space: ∇F = <fx, fy, fz>

Related Questions

How do we find the gradient?

To find the gradient of a function, you calculate the partial derivatives of the function with respect to each variable. For a function ( f(x, y) ), the gradient is represented as a vector ( \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) ). This vector points in the direction of the steepest ascent of the function and its magnitude indicates the rate of increase. You can compute the gradient using calculus techniques, such as differentiation.


What is a gradient function?

Assume you want to know what is the formula of the gradient of the function in multivariable calculus. Let F be a scalar field function in n-dimension. Then, the gradient of a function is: ∇F = <fx1 , fx2, ... , fxn> In the 3-dimensional Cartesian space: ∇F = <fx, fy, fz>


What is the formula for gradient?

Assume you want to know what is the formula of the gradient of the function in multivariable calculus. Let F be a scalar field function in n-dimension. Then, the gradient of a function is: ∇F = <fx1 , fx2, ... , fxn> In the 3-dimensional Cartesian space: ∇F = <fx, fy, fz>


What is gradient formula?

Assume you want to know what is the formula of the gradient of the function in multivariable calculus. Let F be a scalar field function in n-dimension. Then, the gradient of a function is: ∇F = <fx1 , fx2, ... , fxn> In the 3-dimensional Cartesian space: ∇F = <fx, fy, fz>


What is the formula of gradient?

Assume you want to know what is the formula of the gradient of the function in multivariable calculus. Let F be a scalar field function in n-dimension. Then, the gradient of a function is: ∇F = <fx1 , fx2, ... , fxn> In the 3-dimensional Cartesian space: ∇F = <fx, fy, fz>


How do you find a gradient in a equation?

To find the gradient of an equation, you typically take the derivative of the function with respect to its variable. For a function ( y = f(x) ), the gradient (or slope) at any point is given by ( f'(x) ). If the equation is in the form ( Ax + By + C = 0 ), you can rearrange it to the slope-intercept form ( y = mx + b ), where ( m ) represents the gradient. This value indicates how steeply the line rises or falls as you move along the x-axis.


What is the formula for functions?

Assume you want to know what is the formula of the gradient of the function in multivariable calculus. Let F be a scalar field function in n-dimension. Then, the gradient of a function is: ∇F = <fx1 , fx2, ... , fxn> In the 3-dimensional Cartesian space: ∇F = <fx, fy, fz>


What does gradient mean in maths?

In mathematics, particularly in calculus and vector analysis, the gradient refers to a multi-variable generalization of the derivative. It represents the rate and direction of change of a scalar field, typically a function of several variables. The gradient is a vector that points in the direction of the steepest ascent of the function, and its magnitude indicates the rate of increase. Mathematically, for a function ( f(x, y, z) ), the gradient is denoted as ( \nabla f ) and is calculated as the vector of partial derivatives: ( \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) ).


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Why you differentiate problems in maths?

Firstly, and most simply, it may lead to another part of a problem or question or context.Secondly, and importantlyest, (i just invented that word it means most importantly) It can be used to find the gradient of a curve.As you may know, the gradient of a straight line is constanty=mx+c the gradient is mBut for a curve, the gradient is always changing.look at a graph of y=x2 and you will see that an infinite tangents make up the curve each with a different value for m.In short, when you differentiate a function or equation, you get the gradient function, which allows you to find the gradient at any point on the graph y=f(x)differentiate y=x2 (which is the same as find dy/dx)dy/dx=2xso the gradient on the curve y=x2 always 2 times the x value in question. At x=1 the gradient is 2. At x=2 the gradient is 4.In mechanics, if you draw a graph of displacement against time for a moving object, the GRADIENT is equal to the function of velocity. Plot velocity against time and the gradient is equal to Acceleration against time.In any circumstance where a curve is involved differentiation is needed to ind a gradient.Phew, hope that helps and answers your question


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