Vector field notation help

• JaysFan31
In summary, the conversation is about finding the gradient of a vector field with a given notation and simplifying it to solve for the flux across the surface of a sphere. The conversation also involves discussing the meaning of the gradient and its application to scalar functions.

JaysFan31

Just a quick question about notation.

I was given the vector field

F = r + grad(1/bar(r)) where r= (x)i+(y)j+(z)k.
grad is just written as the upside down delta (gradient) and the bar I wrote in the above equation looks like an absolute value around just the r (although I don't know if it is absolute value). Basically I want to find the gradient of (1/bar(r)).

What would be a simplification of this vector field so that I can solve the rest of the problem?

I want to find its flux across the surface of a sphere.

I think F would be ((x^3)-1)/x^2+((y^3)-1)/y^2+((z^3-1)/z^2, but I'm not sure.

$$|\vec{r}|$$ is the modulus, or magnitude, of vector $$\vec{r}$$.

If $\vec{r}= x\vec{i}+ y\vec{j}+ z\vec{k}$ then
$$\frac{1}{||\vec{r}||}= \frac{1}{\sqrt{x^2+y^2+ z^2}}= (x^2+y^2+ z^2)^{-\frac{1}{2}}$$
What is the gradient of that function?

How can you take the gradient of the function if it doesn't have i, j, and k?

Is it 0?

JaysFan31 said:
How can you take the gradient of the function if it doesn't have i, j, and k?

Is it 0?

A better question would be, how could you take the grad of the function if it *did* have i, j, and k? Remember, the gradient acts on a scalar.

JaysFan31 said:
How can you take the gradient of the function if it doesn't have i, j, and k?

Is it 0?
Well, you would first have to know what "gradient" actually means!

Given a function f(x,y,z), how would YOU define
$\nabla f$?

1. What is vector field notation?

Vector field notation is a way of representing vector fields, which are mathematical objects that describe the direction and magnitude of a physical quantity at every point in space.

2. How is vector field notation written?

Vector field notation is typically written using the notation F(x,y,z) = (P(x,y,z), Q(x,y,z), R(x,y,z)), where P, Q, and R are the components of the vector field in the x, y, and z directions, respectively.

3. What are the benefits of using vector field notation?

Vector field notation allows for a compact and concise representation of vector fields, making it easier to visualize and manipulate these mathematical objects. It also allows for easier communication and comparison of vector fields between different scientists and researchers.

4. How is vector field notation used in scientific research?

Vector field notation is used in a variety of scientific fields, including physics, engineering, and mathematics. It is commonly used to describe physical phenomena such as fluid flow, electromagnetic fields, and gravitational fields. It is also used in computer simulations and mathematical models to study complex systems.

5. Are there any common mistakes to avoid when using vector field notation?

One common mistake to avoid when using vector field notation is to mix up the order of the components. It is important to remember that the order of the components corresponds to the x, y, and z directions, respectively. Another mistake to avoid is using a different notation for different vector fields, as this can lead to confusion and errors in calculations.