Gradient:
1. Find the gradient of the function f : x^2y/ z at the point M(-2;3;4).
What is the directional derivative of this function in the direction of vector
I =(12;-4;3)?

2. Find the gradient of the scalar function f = cos(|r|^n) , Where
r : (X, y, Z).
Write the result as a function of r

3. Find the normal line to the level curve of the surface given by
X^2 + 3y^2/2 = 7 at point M(2;-1).

Divergence:
4. Calculate the divergence of the vector Q : (X^2, y^3, XZ^3) at the
field:point (A1,1,3).

Curl:
5. Calculate the curl of the vector field: a = (Z^2, X^2, y^2) at the point
A(1,2,3)

6. Calculate curl(r/r^n)

Use of the "nabla" symbol (V):
7. Find a gradient of scalar product of two vector fields F(x,y,z) and
G(x,y,Z).

Solution Preview

Two things:
1. Here I use the standard vector notations where e_x is a unit vector in the x direction, e_y is a unit vector in the y direction and e_z ...

Solution Summary

The expert examines gradient, divergences and curls. The normal line to the level curve of the surface is determined.

Real Analysis Gradient, Divergence and Curl (II). ... This solution is comprised of a detailed explanation of the problems of Gradient, Divergence and Curl. ...

Real Analysis Gradient, Divergence and Curl (I). ... This solution is comprised of a detailed explanation of the problems of Gradient, Divergence and Curl. ...

... of both sides: (1.7) By the property of the curl and divergence operators we get ... the curl of equation (1.6): (1.11) We know that a curl of a gradient is zero ...

... we need to find the function whose gradient vector is. ... Theorem of Line Integrals, Green's Theorem, Divergence Theorem and ... 1. Find the curl of the vector field F ...

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... 15. Calculate the divergence and curl of the ... So, by partially differentiating with respect to x, y and z, we can calculate the gradient vectors to be: ...