In the early Qing-Dynasty, Jesuit missionaries at the imperial court made western-style cannons to improve and modernize the imperial Chinese army. Known is Ferdinand Verbiest SJ (d. 1688). But where in China have those cannons been made? And where were they used? Only in fortresses? Or did the imperial army have regular artillery-troops like the European armies had in the 17th century? And did the Jesuits not only cast guns but also did research or even wrote books about metallurgy and/or ballistics? And where have those guns gone? Since guns are made of durable materials, some of them should have survived to our days.
Thanks to everybody for some hints.

In fact, in the late Ming the Jesuits were already cooperating with Christian Ming officials (notably Xu Guangqi) and Portuguese artillerymen from Macau to produce European-style cannons for the Ming army and also train Ming soldiers in their use. These cannons were known as Hongyi Pao, i.e. "red[-haired] barbarian cannons", and the first Qing artillery units were made up of surrendered or defected Ming artillerymen along with their Hongyi Pao. After the Qing conquest the Jesuits transferred their services to the Qing emperors.

There is an article by Ma Chujian, "The Introduction of Artillery by the Jesuit Missionaries", in a conference volume entitled Martino Martini: A Humanist and Scientist in Seventeenth-century China (Trento, 1996) that may be useful to you if you can find the book.

I think a bronze or brass field gun, Quing. dated 1693, is in the Weat Point Historical Museum.

I also traced already 3 western-style Chinese cannons, dated 1689, in Europe, 2 in Hungary and 1 in Germany. It would be interesting, how these cannons came to these places outside China an when.
And also interesting for me: where in China and who did make these cannons?

would the missionaries need their government's permission to teach the Chinese such things though?

It almost seems like treason, since there's the possiblity they'd use the cannons against foreign powers

By government do you mean the Holy See or the country of origin - Spain?
If it is the former, I think the Catholic world at the time would be more worried about the rise of Protestantism than China having their lastest technology,winning favour of Chinese government for Catholicism doesnt seem to be a bad option.
If it is the latter, the Society of Jesus seems more like multi-national organisation and as religious organsation it answers more to God and pope than the Spanish kings/queens.

My memory is a bit rusty after reading Paul Kennedy's "The Rise and Fall of the Great Power", as far as I remember part of Paul Kennedy's point is that in Europe there were many centres of production where weapon's concerned, this made a complete domination of Europe for any power impossible and in turn this made the monopoly of weaponry impossible.
Therefore IMO it seems impossible for the Spanish king to charge the Jesuits of treason, in part due to the nature of such organisation, and also due to the lack of capacity to do so,much more its futility. Even if Jesuits didnt impart the technology others might do so to win favour of the local powers,like the introduction of "teppo" by the Portguese and Dutch trader to Japan during Sengoku jidai.