Vector Identity (del operator)

racnna · 2012-07-19T16:22:50+00:00

i am completely lost as to how to go from p^ \frac{1}{m}∇p to \frac{m}{m+1} ∇p^\frac{m+1}{m}

Thread starter racnna
Start date Jul 19, 2012
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Identity Operator Vector Vector identity

In summary, the del operator, represented as ∇ (nabla), is a vector operator used in vector calculus that allows for the expression of physical laws and equations in terms of vectors. It is used to perform operations such as finding the gradient, divergence, and curl of a vector field, as well as solving differential equations and expressing physical laws in terms of vectors. It is a powerful tool for solving problems in physics and engineering.

Jul 19, 2012

racnna

i am completely lost as to how to go from
[tex]p^ \frac{1}{m}∇p[/tex]
to
[tex]\frac{m}{m+1} ∇p^\frac{m+1}{m}[/tex]

Last edited: Jul 19, 2012

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Jul 19, 2012

Muphrid

In general, the chain rule should tell you that,

[tex]\nabla p^a = a p^{a-1} \nabla p[/tex]

Choose an appropriate value for [itex]a[/itex] and see what happens.

Jul 19, 2012

racnna

oh wow...nice trick

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1. What is the del operator and why is it important in vector calculus?

The del operator, symbolized as ∇ (nabla), is a mathematical vector operator used in vector calculus. It is important because it allows us to express physical laws and equations in terms of vectors and is a powerful tool for solving problems in physics and engineering.

2. What is the difference between the gradient, divergence, and curl of a vector field?

The gradient, divergence, and curl are all operations performed by the del operator on a vector field. The gradient measures the rate of change of a scalar quantity in the direction of steepest increase. The divergence measures the tendency of a vector field to spread out or converge at a particular point. The curl measures the rotation or circulation of a vector field around a particular point.

3. How do you use the del operator to find the directional derivative of a scalar field?

The directional derivative of a scalar field is the rate of change of the scalar quantity in a particular direction. To find the directional derivative, we use the dot product between the gradient of the scalar field and the unit vector in the desired direction. In mathematical notation, it can be written as ∇f · ˆv, where ∇f is the gradient of the scalar field f and ˆv is the unit vector in the desired direction.

4. What is the Laplacian operator and how is it related to the del operator?

The Laplacian operator, symbolized as ∇², is a second-order differential operator that is used to measure the rate of change of a scalar field at a specific point. It is related to the del operator by the expression ∇² = ∇ · ∇, where ∇ · ∇ is the divergence of the gradient of the scalar field.

5. How is the del operator used in vector calculus to solve physical problems?

The del operator is used in many different ways to solve physical problems in vector calculus. It can be used to calculate the gradient, divergence, and curl of a vector field, which are important quantities in electromagnetism, fluid mechanics, and other branches of physics. It can also be used to solve differential equations and express physical laws in terms of vectors, making it a powerful tool for solving problems in physics and engineering.