Introduction of gradient in math:
In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change.A generalization of the gradient for functions on a Euclidean space which have values in another Euclidean space is the Jacobian.
Example problem for gradient in math:-
Problem :-
The three variables function is x axis, y axis and z axis the region of three dimensional space it is normally show a f(x,y,z) these is called as scalar field and P is a point and specified direction is no need for the x axis, y axis and z axis. You need to calculate and any change in f is called as directional derivative.
Solution:-
Well, let's start by letting R=x0i+y0j+z0k be the position vector for P. Let the specified direction that we want to move from P be given by the unit vector u = u1i + u2j + u3k. Let Q=(x + x, y + y, z + z) be a point along with the vector in a specified direction. Let Deltas be the scalar value such that vecPQ = Deltashatu, that is s is the length of vecPQ. In following formula for gradient in math
`vecPQ = Deltaxi+ Deltayj+ Deltazk = Deltasu_1i+ Deltasu_2j+ Deltasu_3k`
Let `deltaf ` = f(Q) - f(P). By linear approximation,
`deltaf = fx(P)delta x + fy(P)delta y + fz(P)delta z + erfx(Q)delta x + erfy(Q)delta y + erfz(Q)delta z`
`deltaf = fx(P)(delta s)u1 + fy(P)(delta s)u2 + fz(P)(delta s)u3 + erfx(Q)(delta s)u1 + erfy(Q)(delta s)u2 + erfz(Q)(delta s)u3`
Dividing by deltas, we have,
`(Deltaf)/(deltas) = (delf)/(delx)(p)u_1+ (delf)/(dely)(p)u_2+ (delf)/(delz)(p)u_3+ erf_x(Q)u_1+ erf_y(Q)u_2+ erf_z(Q)u_3`
Two points Q and P are approaches to line. It contains three functions error and finally we got a zero and will get directional derivative.
` (Deltaf)/(deltas) (p)= (delf)/(delx)(p)u_1+ (delf)/(dely)(p)u_2+ (delf)/(delz)(p)u_3`
Or, the dot product,
`gradf(p).hatu`
where gradf is a special function defined as follows,
`(Deltaf) = (delf)/(delx) i+ (delf)/(dely) j+ (delf)/(delz) k`
More texts are skips the vector arrow in given arrow of `gradf ` and `Delta ` symbol is called the Del operator. To writing a symbol as vector it is more helpful for gradient of function produces a vector.
Gradient in math Exercises:
In following math gradient to demonstrate the similarity to differentiation
`grad(f+g) = grad f + grad g`
`grad(f+g) = f grad g + g grad`
`grad(f/g) = f grad g - g grad f /g^2`
`gradf^n = nf^(n-1) grad f `
In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change.A generalization of the gradient for functions on a Euclidean space which have values in another Euclidean space is the Jacobian.
Example problem for gradient in math:-
Problem :-
The three variables function is x axis, y axis and z axis the region of three dimensional space it is normally show a f(x,y,z) these is called as scalar field and P is a point and specified direction is no need for the x axis, y axis and z axis. You need to calculate and any change in f is called as directional derivative.
Solution:-
Well, let's start by letting R=x0i+y0j+z0k be the position vector for P. Let the specified direction that we want to move from P be given by the unit vector u = u1i + u2j + u3k. Let Q=(x + x, y + y, z + z) be a point along with the vector in a specified direction. Let Deltas be the scalar value such that vecPQ = Deltashatu, that is s is the length of vecPQ. In following formula for gradient in math
`vecPQ = Deltaxi+ Deltayj+ Deltazk = Deltasu_1i+ Deltasu_2j+ Deltasu_3k`
Let `deltaf ` = f(Q) - f(P). By linear approximation,
`deltaf = fx(P)delta x + fy(P)delta y + fz(P)delta z + erfx(Q)delta x + erfy(Q)delta y + erfz(Q)delta z`
`deltaf = fx(P)(delta s)u1 + fy(P)(delta s)u2 + fz(P)(delta s)u3 + erfx(Q)(delta s)u1 + erfy(Q)(delta s)u2 + erfz(Q)(delta s)u3`
Dividing by deltas, we have,
`(Deltaf)/(deltas) = (delf)/(delx)(p)u_1+ (delf)/(dely)(p)u_2+ (delf)/(delz)(p)u_3+ erf_x(Q)u_1+ erf_y(Q)u_2+ erf_z(Q)u_3`
Two points Q and P are approaches to line. It contains three functions error and finally we got a zero and will get directional derivative.
` (Deltaf)/(deltas) (p)= (delf)/(delx)(p)u_1+ (delf)/(dely)(p)u_2+ (delf)/(delz)(p)u_3`
Or, the dot product,
`gradf(p).hatu`
where gradf is a special function defined as follows,
`(Deltaf) = (delf)/(delx) i+ (delf)/(dely) j+ (delf)/(delz) k`
More texts are skips the vector arrow in given arrow of `gradf ` and `Delta ` symbol is called the Del operator. To writing a symbol as vector it is more helpful for gradient of function produces a vector.
Gradient in math Exercises:
In following math gradient to demonstrate the similarity to differentiation
`grad(f+g) = grad f + grad g`
`grad(f+g) = f grad g + g grad`
`grad(f/g) = f grad g - g grad f /g^2`
`gradf^n = nf^(n-1) grad f `
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