Vector Operations: Solving Gradient, Divergence, Curl and Laplacian Problems

In summary, the given expressions involve vector and scalar fields and operations such as gradient, divergence, and curl. These expressions can be evaluated step by step using vector identities and basic multiplication rules. The final expression is 36x^2-6xy-12yz+6y+18xz.
  • #1
Erbil
57
0

Homework Statement



$$ \overrightarrow {F}=3xz^{2}i+2xyj-x^{2}k $$ $$\phi =3x^{2}-yz $$ are given vector and scalar fields, respectively.

a) $$\quad \operatorname{div}\left( \operatorname{grad}\phi \operatorname{div}\overrightarrow {F}\right) =\quad? $$

b) $$\quad \operatorname{curl}\left( \phi F\right) =\quad? $$

c) $$\quad \operatorname{div}\left( \phi F\right) =\quad? $$

d) $$\quad \overrightarrow {\nabla }\cdot \left( \nabla \phi \times \overrightarrow {F}\right) =\quad? $$

e) $$\quad \nabla \cdot \left( \overrightarrow {F}\nabla \phi \right) =\quad? $$

I know the operations such as the Gradient, Divergence, Curl, and Laplacian. But I don't have an idea what can I do in this kind of problems?



Homework Equations


http://en.wikipedia.org/wiki/Vector_calculus_identities


The Attempt at a Solution



a)I found gradient of scalar field and divergence of vector field.Gradient of scalar field is a vector and divergence of vector fields is a scalar.So how can I take the divergence of this?
No idea for others.I think I have to use some formula for calculate these..But which? I don't want to a solution just I want to understand logic.Please help me to figure this:) Then I will try to do it myself.

And also,I'd like to say that I'm sorry for my bad English.
 
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  • #2
Gradient of scalar field is a vector and divergence of vector fields is a scalar.So how can I take the divergence of this?
Vector*Scalar (better: written as scalar*vector) is a vector again, and has a divergence.

No idea for others.I think I have to use some formula for calculate these.
Vector*Vector: Scalar product ("dot product")
Vector x Vector: Cross-product
Scalar*Vector or Vector*Scalar: Scalar multiplication of the vector
Scalar*Scalar: Just like multiplication with real numbers

You can evaluate these expressions step by step.
 
  • #3
@mfb; Is there any another way,because there are a lot of process with step by step? Now,I'm trying step by step.I will tell the result,but I don't know where can I control it?
 
  • #4
WolframAlpha and other computer algebra systems should be able to do that.
There are many steps, but they are all easy, and you can even re-use some.

For more complicated expressions, vector identities can be useful, but I think you cannot use them (in a meaningful way) here.
 
  • #5
mfb said:
WolframAlpha and other computer algebra systems should be able to do that.
There are many steps, but they are all easy, and you can even re-use some.

For more complicated expressions, vector identities can be useful, but I think you cannot use them (in a meaningful way) here.

OK.Here is my expression.

= (18z^2-12+72x+2z-6z^2+2y-6yz) + ( 24-4z-3z^2-8y+2x-6xz) + (36xz+36x^2-9z^2-4y+2x-12xz-6yz-6xy)

= 36x^2-6xy-12yz+6y+18xz am I right?
 

FAQ: Vector Operations: Solving Gradient, Divergence, Curl and Laplacian Problems

1. What are vector operations?

Vector operations refer to mathematical operations performed on vectors, which are quantities that have both magnitude and direction. These operations include addition, subtraction, multiplication, and division of vectors.

2. What is the gradient of a vector field?

The gradient of a vector field is a vector that points in the direction of the greatest increase of the field and has a magnitude equal to the rate of change of the field in that direction. It is commonly used to calculate the slope of a surface or the direction of flow in a fluid.

3. How is the divergence of a vector field calculated?

The divergence of a vector field is calculated by taking the dot product of the vector field with the del operator (∇). This operation results in a scalar value that represents the amount of "outwardness" or "inwardness" of the vector field at a given point.

4. What does the curl of a vector field represent?

The curl of a vector field is a vector that represents the amount of rotation or circulation of the field at a given point. It is calculated by taking the cross product of the vector field with the del operator (∇) and has applications in fluid dynamics and electromagnetism.

5. What is the Laplacian operator used for?

The Laplacian operator (∇2) is used to calculate the rate of change of a scalar field at a given point. It is commonly used in physics and engineering to solve differential equations and analyze the behavior of physical systems.

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