# How to use ∂ function in my code or how to compute the mixed second-order partial derivatives or first-order cross partial derivatives of the velocity field \*\*u\*\*?
**URL:** https://discourse.julialang.org/t/how-to-use-function-in-my-code-or-how-to-compute-the-mixed-second-order-partial-derivatives-or-first-order-cross-partial-derivatives-of-the-velocity-field-u/134213
**Category:** General Usage
**Tags:** question
**Created:** [November 29, 2025, 6:44am UTC](https://discourse.julialang.org/t/how-to-use-function-in-my-code-or-how-to-compute-the-mixed-second-order-partial-derivatives-or-first-order-cross-partial-derivatives-of-the-velocity-field-u/134213 "2025-11-29T06:44:38Z")
**Posts on this page:** 7
**Page:** 1
### Author:  [@Mr.Fool](https://discourse.julialang.org/u/Mr.Fool)
#### Post date: [November 29, 2025, 6:44am UTC](https://discourse.julialang.org/t/how-to-use-function-in-my-code-or-how-to-compute-the-mixed-second-order-partial-derivatives-or-first-order-cross-partial-derivatives-of-the-velocity-field-u/134213/1 "2025-11-29T06:44:38Z")
```julia-auto
using WaterLily,StaticArrays,GLMakie,Makie,Plots
using LinearAlgebra: eigvals, Hermitian
function donut(L;Re=1e3,mem=Array,U=1)
center,r = SA[3L,3L,0], L/2
bodya = AutoBody() do xyz,t
x,y,z = xyz - center
√sum(abs2,SA[x,y,0])-r
end
cycenter=SA[3L,3L,4L]
bodyb = AutoBody() do xyz,t
x,y,z = xyz - cycenter
a=√sum(abs2,SA[x,y,0])-L
b=√sum(abs2,SA[0,0,z])-r
max(a,b)
end
body=bodya+bodyb
Simulation((8L,6L,8L),(U,0,0),L;ν=U*L/Re,body,mem)
end
function Q(I::CartesianIndex{3}, u)
J = @SMatrix [∂(i, j, I, u) for i in 1:3, j in 1:3]
S = (J + J') / 2
Ω = (J - J') / 2
norm_S_sq = sum(S.^2)
norm_Ω_sq = sum(Ω.^2)
0.5 * (norm_Ω_sq - norm_S_sq)
end
function ω!(arr, sim)
a = sim.flow.σ
WaterLily.@inside a[I] = Q(I,sim.flow.u)
copyto!(arr, a[inside(a)]) # copy to CPU
end
sim = donut(2^3);
t₀ = sim_time(sim)
duration = 100.0
step = 0.25
viz!(sim;f=ω!,duration,step,framerate=5,algorithm=:mip,colormap=:algae)
```
The code above is what I want to achieve. I saw the following code in the StaticArrays package.
```julia-auto
function λ₂(I::CartesianIndex{3},u)
J = @SMatrix [∂(i,j,I,u) for i ∈ 1:3, j ∈ 1:3]
S,Ω = (J+J')/2,(J-J')/2
eigvals(Hermitian(S^2+Ω^2))[2]
end
```
So I want to write a function following this example. But when I run my code, an error occurs. The error is as follows.
```julia-auto
1-element ExceptionStack:
LoadError: TaskFailedException
nested task error: UndefVarError: `∂` not defined in `Main`
Suggestion: check for spelling errors or missing imports.
Stacktrace:
[1] (::var"#Q##2#Q##3"{CartesianIndex{3}, Array{Float32, 4}})(i::Int64, j::Int64)
@ Main D:\julia\julia_pakage\packages\StaticArrays\DsPgf\src\SMatrix.jl:63
[2] macro expansion
@ D:\julia\julia_pakage\packages\StaticArrays\DsPgf\src\SMatrix.jl:64 [inlined]
[3] Q(I::CartesianIndex{3}, u::Array{Float32, 4})
@ Main f:\julia_code\3D阶梯圆柱\threed.jl:36
[4] macro expansion
@ D:\julia\julia_pakage\packages\WaterLily\O8g3K\src\util.jl:146 [inlined]
[5] macro expansion
@ D:\julia\julia_pakage\packages\KernelAbstractions\X5fk1\src\macros.jl:314 [inlined]
[6] (::var"#cpu_##kern_#286#ω!##5")( __ctx__ ::KernelAbstractions.CompilerMetadata{KernelAbstractions.NDIteration.DynamicSize, KernelAbstractions.NDIteration.NoDynamicCheck, CartesianIndex{3}, CartesianIndices{3, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}, Base.OneTo{Int64}}}, KernelAbstractions.NDIteration.NDRange{3, KernelAbstractions.NDIteration.DynamicSize, KernelAbstractions.NDIteration.StaticSize{(64, 1, 1)}, CartesianIndices{3, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}, Base.OneTo{Int64}}}, Nothing}}, a::Array{Float32, 3}, u::Array{Float32, 4}, I0::CartesianIndex{3})
@ Main .\none:0
[7] __thread_run(tid::Int64, len::Int64, rem::Int64, obj::KernelAbstractions.Kernel{KernelAbstractions.CPU, KernelAbstractions.NDIteration.StaticSize{(64,)}, KernelAbstractions.NDIteration.DynamicSize, var"#cpu_##kern_#286#ω!##5"}, ndrange::Tuple{Int64, Int64, Int64}, iterspace::KernelAbstractions.NDIteration.NDRange{3, KernelAbstractions.NDIteration.DynamicSize, KernelAbstractions.NDIteration.StaticSize{(64, 1, 1)}, CartesianIndices{3, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}, Base.OneTo{Int64}}}, Nothing}, args::Tuple{Array{Float32, 3}, Array{Float32, 4}, CartesianIndex{3}}, dynamic::KernelAbstractions.NDIteration.NoDynamicCheck)
@ KernelAbstractions D:\julia\julia_pakage\packages\KernelAbstractions\X5fk1\src\cpu.jl:145
[8] (::KernelAbstractions.var"# __run##4#__ run##5"{KernelAbstractions.Kernel{KernelAbstractions.CPU, KernelAbstractions.NDIteration.StaticSize{(64,)}, KernelAbstractions.NDIteration.DynamicSize, var"#cpu_##kern_#286#ω!##5"}, Tuple{Int64, Int64, Int64}, KernelAbstractions.NDIteration.NDRange{3, KernelAbstractions.NDIteration.DynamicSize, KernelAbstractions.NDIteration.StaticSize{(64, 1, 1)}, CartesianIndices{3, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}, Base.OneTo{Int64}}}, Nothing}, Tuple{Array{Float32, 3}, Array{Float32, 4}, CartesianIndex{3}}, KernelAbstractions.NDIteration.NoDynamicCheck, Int64})()
@ KernelAbstractions D:\julia\julia_pakage\packages\KernelAbstractions\X5fk1\src\cpu.jl:120
...and 19 more exceptions.
Stacktrace:
[1] sync_end(c::Channel{Any})
@ Base .\task.jl:601
[2] macro expansion
@ .\task.jl:634 [inlined]
[3] __run(obj::KernelAbstractions.Kernel{KernelAbstractions.CPU, KernelAbstractions.NDIteration.StaticSize{(64,)}, KernelAbstractions.NDIteration.DynamicSize, var"#cpu_##kern_#286#ω!##5"}, ndrange::Tuple{Int64, Int64, Int64}, iterspace::KernelAbstractions.NDIteration.NDRange{3, KernelAbstractions.NDIteration.DynamicSize, KernelAbstractions.NDIteration.StaticSize{(64, 1, 1)}, CartesianIndices{3, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}, Base.OneTo{Int64}}}, Nothing}, args::Tuple{Array{Float32, 3}, Array{Float32, 4}, CartesianIndex{3}}, dynamic::KernelAbstractions.NDIteration.NoDynamicCheck, static_threads::Bool)
@ KernelAbstractions D:\julia\julia_pakage\packages\KernelAbstractions\X5fk1\src\cpu.jl:119
[4] (::KernelAbstractions.Kernel{KernelAbstractions.CPU, KernelAbstractions.NDIteration.StaticSize{(64,)}, KernelAbstractions.NDIteration.DynamicSize, var"#cpu_##kern_#286#ω!##5"})(::Array{Float32, 3}, ::Vararg{Any}; ndrange::Tuple{Int64, Int64, Int64}, workgroupsize::Nothing)
@ KernelAbstractions D:\julia\julia_pakage\packages\KernelAbstractions\X5fk1\src\cpu.jl:46
[5] (::var"##kern#285#ω!##9"{Simulation})(a::Array{Float32, 3}, u::Array{Float32, 4})
@ Main D:\julia\julia_pakage\packages\WaterLily\O8g3K\src\util.jl:149
[6] macro expansion
@ D:\julia\julia_pakage\packages\WaterLily\O8g3K\src\util.jl:151 [inlined]
[7] ω!(arr::Array{Float32, 3}, sim::Simulation)
@ Main f:\julia_code\3D阶梯圆柱\threed.jl:56
[8] viz!(sim::Simulation; f::typeof(ω!), duration::Float64, step::Float64, remeasure::Bool, verbose::Bool, λ::Function, udf::Nothing, udf_kwargs::Nothing, d::Int64, CIs::Nothing, cut::Nothing, tidy_colormap::Bool, body::Bool, body_color::Symbol, body2mesh::Bool, video::Nothing, hidedecorations::Bool, elevation::Float64, azimuth::Float64, framerate::Int64, compression::Int64, theme::Nothing, fig_size::Nothing, fig_pad::Int64, fig::Nothing, ax::Nothing, kwargs::@Kwargs{algorithm::Symbol, colormap::Symbol})
@ WaterLilyMakieExt D:\julia\julia_pakage\packages\WaterLily\O8g3K\ext\WaterLilyMakieExt.jl:184
[9] top-level scope
@ f:\julia_code\3D阶梯圆柱\threed.jl:66
[10] eval(m::Module, e::Any)
@ Core .\boot.jl:489
[11] include_string(mapexpr::typeof(identity), mod::Module, code::String, filename::String)
@ Base .\loading.jl:2842
[12] include_string(m::Module, txt::String, fname::String)
@ Base .\loading.jl:2852
[13] inlineeval(m::Module, code::String, code_line::Int64, code_column::Int64, file::String; softscope::Bool)
@ VSCodeServer c:\Users\intel\.vscode\extensions\julialang.language-julia-1.158.2\scripts\packages\VSCodeServer\src\eval.jl:271
[14] (::VSCodeServer.var"#repl_runcode_request##6#repl_runcode_request##7"{Bool, Bool, Bool, Module, String, Int64, Int64, String, VSCodeServer.ReplRunCodeRequestParams})()
@ VSCodeServer c:\Users\intel\.vscode\extensions\julialang.language-julia-1.158.2\scripts\packages\VSCodeServer\src\eval.jl:181
[15] withpath(f::VSCodeServer.var"#repl_runcode_request##6#repl_runcode_request##7"{Bool, Bool, Bool, Module, String, Int64, Int64, String, VSCodeServer.ReplRunCodeRequestParams}, path::String)
@ VSCodeServer c:\Users\intel\.vscode\extensions\julialang.language-julia-1.158.2\scripts\packages\VSCodeServer\src\repl.jl:278
[16] (::VSCodeServer.var"#repl_runcode_request##4#repl_runcode_request##5"{Bool, Bool, Bool, Module, String, Int64, Int64, String, VSCodeServer.ReplRunCodeRequestParams})()
@ VSCodeServer c:\Users\intel\.vscode\extensions\julialang.language-julia-1.158.2\scripts\packages\VSCodeServer\src\eval.jl:179
[17] hideprompt(f::VSCodeServer.var"#repl_runcode_request##4#repl_runcode_request##5"{Bool, Bool, Bool, Module, String, Int64, Int64, String, VSCodeServer.ReplRunCodeRequestParams})
@ VSCodeServer c:\Users\intel\.vscode\extensions\julialang.language-julia-1.158.2\scripts\packages\VSCodeServer\src\repl.jl:38
[18] #repl_runcode_request##2
@ c:\Users\intel\.vscode\extensions\julialang.language-julia-1.158.2\scripts\packages\VSCodeServer\src\eval.jl:150 [inlined]
[19] with_logstate(f::VSCodeServer.var"#repl_runcode_request##2#repl_runcode_request##3"{Bool, Bool, Bool, Module, String, Int64, Int64, String, VSCodeServer.ReplRunCodeRequestParams}, logstate::Base.CoreLogging.LogState)
@ Base.CoreLogging .\logging\logging.jl:540
[20] with_logger
@ .\logging\logging.jl:651 [inlined]
[21] (::VSCodeServer.var"#repl_runcode_request##0#repl_runcode_request##1"{VSCodeServer.ReplRunCodeRequestParams})()
@ VSCodeServer c:\Users\intel\.vscode\extensions\julialang.language-julia-1.158.2\scripts\packages\VSCodeServer\src\eval.jl:263
[22] (::VSCodeServer.var"#start_eval_backend##0#start_eval_backend##1")()
@ VSCodeServer c:\Users\intel\.vscode\extensions\julialang.language-julia-1.158.2\scripts\packages\VSCodeServer\src\eval.jl:34
in expression starting at f:\julia_code\3D阶梯圆柱\threed.jl:66
```
So I want to konw how can I use ∂ in my code or how can I compute the mixed second-order partial derivatives or first-order cross partial derivatives of the velocity field **u** , using a **central difference scheme**.
I am looking forward to your answer. Thank you very much.
---
### Author:  [@nilshg](https://discourse.julialang.org/u/nilshg)
#### Post date: [November 29, 2025, 6:52am UTC](https://discourse.julialang.org/t/how-to-use-function-in-my-code-or-how-to-compute-the-mixed-second-order-partial-derivatives-or-first-order-cross-partial-derivatives-of-the-velocity-field-u/134213/2 "2025-11-29T06:52:51Z")
Are you doing this as part of a university course or something? The same question was asked yesterday:
> [@How to use partial derivate in my code?](https://discourse.julialang.org/t/how-to-use-partial-derivate-in-my-code/134176/4):
>
> If you just want to know how to type it, the REPL help tells you: help?\> ∂ "∂" can be typed by \partial\ This kind of tab completion is supported in the REPL and can also be configured in most modern editors. But by default, ∂ doesn’t have any definition in Julia, and is just another possible identifier, like x or f. It is up to you to assign a meaning to it.
---
### Author:  [@Mr.Fool](https://discourse.julialang.org/u/Mr.Fool)
#### Post date: [November 29, 2025, 7:00am UTC](https://discourse.julialang.org/t/how-to-use-function-in-my-code-or-how-to-compute-the-mixed-second-order-partial-derivatives-or-first-order-cross-partial-derivatives-of-the-velocity-field-u/134213/3 "2025-11-29T07:00:33Z")
I use it to visualize three-dimensional vorticity fields, which constitutes an important part of my scientific research.
---
### Author:  [@gdalle](https://discourse.julialang.org/u/gdalle)
#### Post date: [November 29, 2025, 7:25am UTC](https://discourse.julialang.org/t/how-to-use-function-in-my-code-or-how-to-compute-the-mixed-second-order-partial-derivatives-or-first-order-cross-partial-derivatives-of-the-velocity-field-u/134213/4 "2025-11-29T07:25:51Z")
Where do you expect the \partial function to come from? It isn’t defined anywhere in your code. Usually, these computations are performed with an [automatic differentiation](https://juliadiff.org/) library, so unless you have an explicit formula for `∂(i, j, I, u)`, you may need to pick one of these libraries in the Julia ecosystem.
Also, please try to format your posts using the correct backticks `````, so that the code is displayed properly, see [Please read: make it easier to help you](https://discourse.julialang.org/t/please-read-make-it-easier-to-help-you/14757).
---
### Author:  [@Mr.Fool](https://discourse.julialang.org/u/Mr.Fool)
#### Post date: [November 29, 2025, 7:48am UTC](https://discourse.julialang.org/t/how-to-use-function-in-my-code-or-how-to-compute-the-mixed-second-order-partial-derivatives-or-first-order-cross-partial-derivatives-of-the-velocity-field-u/134213/5 "2025-11-29T07:48:57Z")
Thank you for your suggestion. I found ∂(i, j, I, u) in the path `D:\julia\julia_pakage\packages\WaterLily\08g3K\src\Metrics.jl`, which contains the definition of ∂(i, j, I, u) as follows.
```julia-auto
"""
∂(i,j,I,u)
Compute ``∂uᵢ/∂xⱼ`` at center of cell `I`. Cross terms are computed
less accurately than inline terms because of the staggered grid.
"""
@fastmath @inline (i,j,I,u) = (i==j ? ∂(i,I,u) :
@inbounds(u[I+δ(j,I),i]+u[I+δ(j,I)+δ(i,I),i]
-u[I-δ(j,I),i]-u[I-δ(j,I)+δ(i,I),i])/4)
```
---
### Author:  [@gdalle](https://discourse.julialang.org/u/gdalle)
#### Post date: [November 29, 2025, 7:21pm UTC](https://discourse.julialang.org/t/how-to-use-function-in-my-code-or-how-to-compute-the-mixed-second-order-partial-derivatives-or-first-order-cross-partial-derivatives-of-the-velocity-field-u/134213/6 "2025-11-29T19:21:05Z")
The symbol `∂` is not exported by WaterLily.jl, so you may want to add
```julia-auto
using WaterLily: ∂
```
to your imports
---
### Author:  [@Mr.Fool](https://discourse.julialang.org/u/Mr.Fool)
#### Post date: [November 30, 2025, 1:54am UTC](https://discourse.julialang.org/t/how-to-use-function-in-my-code-or-how-to-compute-the-mixed-second-order-partial-derivatives-or-first-order-cross-partial-derivatives-of-the-velocity-field-u/134213/7 "2025-11-30T01:54:11Z")
Thank you very much for your explanation. I tried it and it works.