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  • Partial integration of dot products and the $\nabla$ operator
    Any help is greatly appreciated, I'm only familiar with very simple calculations with partial integration and the double integrals, dot products and $\nabla$ are confusing me EDIT: 1st part answered, still need help on the second part!
  • multivariable calculus - Understanding the notation $\nabla . . .
    I came across this problem when going over some material related to shear stress vector As far as I know the symbol $\nabla$ has a couple of different meanings Let $\vec {i},\vec {j},\vec {k}$ be th
  • multivariable calculus - Is $\nabla = -\text {div}$? - Mathematics . . .
    Another good reason is then that this definition makes the Laplacian a nonnegative operator since $$ \int_V f \nabla \cdot (\nabla f) = -\int_V \lvert \nabla f \rvert^2 + \text {boundary terms} $$ Really, a text should say somewhere which convention is used
  • Show that $\\int_C d\\mathbf{r} \\times \\mathbf{F} = \\int_S (d . . .
    To answer your first question, yes, there should be an extra negative sign As for your second question, I quote the answer to the linked question and add the extra negative sign: \begin {align} \dots = \displaystyle a_j \int_S (\mathrm dS_i \partial_j F_i -\mathrm d S_j \partial_i F_i) \tag {1} \\ = a \cdot \int_S (\mathrm d S\cdot ( \nabla F) - \mathrm d S ( \nabla \cdot F)) \tag {2
  • How do we get $\nabla \cdot \mathbf {B} = 0$ from $\nabla \times . . .
    Taking the divergence of both sides gets $$ \nabla \cdot (\nabla \times \vec {E}) = \nabla \cdot \frac {\partial \vec {B}} {\partial t}$$ The left hand side is zero This is a vector calculus identitiy that you can check by writing out the derivatives On the right hand side you can rearrange the space and time derivative to get $$ \frac {\partial (\nabla \cdot \vec {B})} {\partial t} = 0$$ If
  • Two different meanings of $\nabla$ with subscript?
    I am trying to understand the meaning of $\\nabla$ when it appears with subscript I have found two separate Physics SE answers that imply different meanings The notation $\\vec \\nabla_B$ means si
  • What does $\nabla \nabla$ mean? (nabla nabla, del del)
    So to avoid any confusion, the Laplacian is donated by $\nabla^2=\nabla\cdot\nabla$, the divergent of the gradient and is a scalar, while the Hessian Matrix is the gradient of the gradient, therefore explicitly expressed as $\nabla\!\nabla$





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