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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?
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 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>
What is a gradient in maths terms?
In mathematical terms, a gradient refers to the rate of change of a function with respect to its variables. In the context of a function of several variables, the gradient is a vector that points in the direction of the steepest ascent and whose magnitude represents the rate of increase in that direction. For a function ( f(x, y) ), the gradient is expressed as ( \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) ). In single-variable calculus, the gradient is simply the derivative of the function.
What is the gradient and wind a set of 7X plus 4Y -10?
To find the gradient of the function ( f(x, y) = 7x + 4y - 10 ), we compute the partial derivatives with respect to ( x ) and ( y ). The gradient, denoted as ( \nabla f ), is given by ( \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) = (7, 4) ). This indicates that the steepest ascent of the function occurs in the direction of the vector ( (7, 4) ). The term "wind" is not standard in this context; if you meant "direction of the gradient," it is indicated by the gradient vector.
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>
What is a gradient in maths terms?
In mathematical terms, a gradient refers to the rate of change of a function with respect to its variables. In the context of a function of several variables, the gradient is a vector that points in the direction of the steepest ascent and whose magnitude represents the rate of increase in that direction. For a function ( f(x, y) ), the gradient is expressed as ( \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) ). In single-variable calculus, the gradient is simply the derivative of the function.
What is the gradient and wind a set of 7X plus 4Y -10?
To find the gradient of the function ( f(x, y) = 7x + 4y - 10 ), we compute the partial derivatives with respect to ( x ) and ( y ). The gradient, denoted as ( \nabla f ), is given by ( \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) = (7, 4) ). This indicates that the steepest ascent of the function occurs in the direction of the vector ( (7, 4) ). The term "wind" is not standard in this context; if you meant "direction of the gradient," it is indicated by the gradient vector.
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) ).
What is a positive gradient in math?
A positive gradient is a characteristic of a function whose value increases as the value of the argument increases. So, if y is a function, f(x), of x, then an increase in the value of x is accompanied by an increase in the value of y.
Why do you use gradient?
The derivative of a function, df/dx, is to single variable calculus as the gradient of a function, ∇f, is to multivariable calculus.If f is a function of three variables, x, y, and z, then the gradient of f is the vector function ∇f(x, y, z) = All of the uses of derivatives in single variable calculus are analogous to the uses of gradients in multivariable calculus:In single variable calculus the derivative tells us the instantaneous rate of change at some point, [x, f(x)]. In multivariable calculus, the gradient of a function tells us the instantaneous rate of change at some point, [x, y, f(x,y)], or if the function is of more than two variables, ∇f would tell us the instantaneous rate of change at a point [x, y, z, ….., f(x, y, z, ….)]. One Important difference in calculus of more than one variable is that a function can have many different rates of changes at one point. To understand why this is so, imagine that you are standing on a hilltop which is defined by a function of two variables f(x, y). The downward slope of the hill, the gradient, is different depending on the direction you look; to find the slope you need to specify a direction. This is why we take the 'directional derivative' which is simply the dot product of the gradient with a unit direction vector (the direction you are looking down the hill). For example suppose we want to find the instantaneous rate of change of the function f(x,y) = x2 + y2 at the point (2,1) in the direction of v = :The directional derivative in the direction of v = ∇f(x, y) ● = < ∂f/∂x(x, y, z), ∂f/∂y(x, y, z)> ● = ● = 2y evaluated at (2,1) = 2.Let's continue our comparison of derivatives and gradients. In single variable calculus a derivative of a function is equal to zero at a maximum or minimum value of the function. This fact can be used in practical applications that require maximizing and minimizing functions of one variable. The same is said of the gradient in multivariable calculus. By setting the gradient of a multivariable function equal to zero, we can solve for the point of maximum or minimum values.
