## Integration of a vector function in spherical coordinates, where the unit vector cannot be represented as the standard radial unit vector?

While trying to find out the electric field due to a charged sphere without using Gauss Law, I encountered an integration $$iiintfrac{r^2sintheta drdtheta dphi}{left|vec{r}-vec{r’}right|^3}left(vec{r}-vec{r’}right)$$
Here, $$vec{r’}$$ is a constant, while $$vec{r}$$ is the radial vector.

The exact part which is troubling me the most is $$int_0^{r_o}frac{r^2left(vec{r}-vec{r’}right)}{left|vec{r}-vec{r’}right|^3}dr$$
or, $$int_0^{r_o}frac{r^2(widehat{vec{r}-vec{r’}})}{left|vec{r}-vec{r’}right|^2}dr$$
Here, the unit vector isn’t $$hat{r}$$, which can be written as $$costhetacosphihat{i}+costhetasinphihat{j}+sinthetahat{k}$$, as I saw in another answer Vector valued integral in spherical coordinates.
Rather, it is $$(widehat{vec{r}-vec{r’}})$$. This cannot be taken out of the integral as the unit vectors of Cartesian coordinates $$hat{i}, hat{j}$$ and $$hat{k}$$, since they are dependent on $$theta$$ and $$phi$$. So, how exactly should I proceed?

## unity – How to create a standard pbr material with shader graph?

I want to create a standard pbr material that is editable with the shader graph, but when I install the package and go the Create -> Shader tab, I only have 2 graph options: Empty Shader Graph, and Sub Graph. Isn’t there a shader graph that starts me off with a basic pbr material with color and reflections? I’m using unity 2020, and the default render pipeline. Do I need to switch to ldrp or hdrp?

## statistics – Question regarding the shape of a probability distribution given a specific condition on it’s standard deviation

Let $$X$$ denote a random variable, with mean $$mu$$ and variance $$sigma^2$$. Suppose the following condition to be true, where

$$begin{equation} P(X=mu+sigma)=P(X=mu-sigma)=0.5 end{equation}$$

Is there any specific shape or probability distribution that would satisfy this?

I believe this is saying that all possible values of $$X$$ lie within one standard deviation, if this is true, what else does it imply about $$X$$?

Thanks in advance for any clarifications.

## documentation – Compare the 830-1984 , 830-1993, 830-1998 to the IEEE Standard in terms of structure, contents, and style. Please Explain it briefly

Compare the 830-1984 , 830-1993, 830-1998 to the IEEE Standard in terms of structure, contents, and style. Please Explain it briefly.
Access and Read the IEEE Std 830-1993, using internet, try to find at least two other templates for requirements specification . Compare the templates to the IEEE Standard in terms of structure, contents, and style.

## audit – What alternative standard for ISO 27001 can be used in Australia?

I am looking for alternatives, that are less strict and less time consuming, than ISO 27001. Australia is in the Commonwealth, so maybe Cyber Essentials Plus could work, but I do not know if that plays a part in it being recognized by the Australian Government. Any suggestions? Thanks!

## plugins – Add Woocommerce Fast and standard delivery optin

plugins – Add Woocommerce Fast and standard delivery optin – WordPress Development Stack Exchange

## generating random pair key has standard?

If a crypto wallet generates all pair key between small range for example between 1-100, the risk and possibility of creating the same key is an increase. Generating random pair keys for cryptocurrency has standard? If the answer is No what the solution cryptocurrency for it?

## Standard Error in Nonliner Model Fit

What does the Standard Error in Non Linear Model Fit signify ? How is it calculated ? How can I interpret the value in terms of goodness of my fit ?