{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:NGUOUV367QEDMIYIP5MAWWGKCY","short_pith_number":"pith:NGUOUV36","canonical_record":{"source":{"id":"2607.01096","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AP","submitted_at":"2026-07-01T15:49:42Z","cross_cats_sorted":[],"title_canon_sha256":"79cb7bb0616ed87178d36bfdc66a615866b4c8d498aa4e886cbae1afa4025b5d","abstract_canon_sha256":"38faa5574e83545871d8db35c01af2fbb46aa9923c82e9548ddf7467e79f6210"},"schema_version":"1.0"},"canonical_sha256":"69a8ea577efc083623087f580b58ca1619d7a4c838a4b1c0bb7f7efad20b740b","source":{"kind":"arxiv","id":"2607.01096","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.01096","created_at":"2026-07-02T01:18:28Z"},{"alias_kind":"arxiv_version","alias_value":"2607.01096v1","created_at":"2026-07-02T01:18:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.01096","created_at":"2026-07-02T01:18:28Z"},{"alias_kind":"pith_short_12","alias_value":"NGUOUV367QED","created_at":"2026-07-02T01:18:28Z"},{"alias_kind":"pith_short_16","alias_value":"NGUOUV367QEDMIYI","created_at":"2026-07-02T01:18:28Z"},{"alias_kind":"pith_short_8","alias_value":"NGUOUV36","created_at":"2026-07-02T01:18:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:NGUOUV367QEDMIYIP5MAWWGKCY","target":"record","payload":{"canonical_record":{"source":{"id":"2607.01096","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AP","submitted_at":"2026-07-01T15:49:42Z","cross_cats_sorted":[],"title_canon_sha256":"79cb7bb0616ed87178d36bfdc66a615866b4c8d498aa4e886cbae1afa4025b5d","abstract_canon_sha256":"38faa5574e83545871d8db35c01af2fbb46aa9923c82e9548ddf7467e79f6210"},"schema_version":"1.0"},"canonical_sha256":"69a8ea577efc083623087f580b58ca1619d7a4c838a4b1c0bb7f7efad20b740b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-02T01:18:28.927366Z","signature_b64":"69fD/KKmVCuI9W7KQ/XLyoNNKCL1Huli6k9O/U7EjyhbSHaRiWj/JXf6eYB5rePicWxFge2vgsYAJS+gAIK9Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"69a8ea577efc083623087f580b58ca1619d7a4c838a4b1c0bb7f7efad20b740b","last_reissued_at":"2026-07-02T01:18:28.926965Z","signature_status":"signed_v1","first_computed_at":"2026-07-02T01:18:28.926965Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.01096","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-02T01:18:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NWJLnHCWdorg9jwr4pFkXwkBlJO4IlsnzmophmN/z+CqCBYmTm1sJ3/jw++WsfvDgT3SQYNWos11Fja7sSkfAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-10T15:11:27.283343Z"},"content_sha256":"0f87002d46cfc24500550b444da6d7bcdc87a9c05568c1d1e2ae5efa718ea469","schema_version":"1.0","event_id":"sha256:0f87002d46cfc24500550b444da6d7bcdc87a9c05568c1d1e2ae5efa718ea469"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:NGUOUV367QEDMIYIP5MAWWGKCY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Explicit formulas for gradients and the divergence in n-dimensional spherical coordinates","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Bern Rummler, Gudrun Th\\\"ater","submitted_at":"2026-07-01T15:49:42Z","abstract_excerpt":"We use the Laplacian in n-dimensional spherical coordinates (n>1) to write the divergence of a vector field defined on radially symmetric domains in the context of vector calculus. We apply straightforward equations of vector calculus with the nabla operator and the transformation matrices from Cartesian to spherical polar coordinates. One needs the divergence of a vector field e.g. to prove that vector fields are eigenfunctions of the Stokes operator on n-dimensional annuli and balls. Our divergence formula in partial derivatives in n-dimensional spherical polar coordinates is an important st"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.01096","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.01096/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-02T01:18:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"svnRJy1NTnx/MfMJ+GEa3EK5zztkDlTvp1ICPEONUZLVCJINe2B6IRHh0JMeAPIxRRCwvcDgxLD7ghoxF/KtAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-10T15:11:27.283744Z"},"content_sha256":"740a5e90bfc392645771e4c4d707eba1deb42f53a4b554db29ce9c8d853d27bd","schema_version":"1.0","event_id":"sha256:740a5e90bfc392645771e4c4d707eba1deb42f53a4b554db29ce9c8d853d27bd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NGUOUV367QEDMIYIP5MAWWGKCY/bundle.json","state_url":"https://pith.science/pith/NGUOUV367QEDMIYIP5MAWWGKCY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NGUOUV367QEDMIYIP5MAWWGKCY/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-07-10T15:11:27Z","links":{"resolver":"https://pith.science/pith/NGUOUV367QEDMIYIP5MAWWGKCY","bundle":"https://pith.science/pith/NGUOUV367QEDMIYIP5MAWWGKCY/bundle.json","state":"https://pith.science/pith/NGUOUV367QEDMIYIP5MAWWGKCY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NGUOUV367QEDMIYIP5MAWWGKCY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:NGUOUV367QEDMIYIP5MAWWGKCY","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"38faa5574e83545871d8db35c01af2fbb46aa9923c82e9548ddf7467e79f6210","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AP","submitted_at":"2026-07-01T15:49:42Z","title_canon_sha256":"79cb7bb0616ed87178d36bfdc66a615866b4c8d498aa4e886cbae1afa4025b5d"},"schema_version":"1.0","source":{"id":"2607.01096","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.01096","created_at":"2026-07-02T01:18:28Z"},{"alias_kind":"arxiv_version","alias_value":"2607.01096v1","created_at":"2026-07-02T01:18:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.01096","created_at":"2026-07-02T01:18:28Z"},{"alias_kind":"pith_short_12","alias_value":"NGUOUV367QED","created_at":"2026-07-02T01:18:28Z"},{"alias_kind":"pith_short_16","alias_value":"NGUOUV367QEDMIYI","created_at":"2026-07-02T01:18:28Z"},{"alias_kind":"pith_short_8","alias_value":"NGUOUV36","created_at":"2026-07-02T01:18:28Z"}],"graph_snapshots":[{"event_id":"sha256:740a5e90bfc392645771e4c4d707eba1deb42f53a4b554db29ce9c8d853d27bd","target":"graph","created_at":"2026-07-02T01:18:28Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2607.01096/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We use the Laplacian in n-dimensional spherical coordinates (n>1) to write the divergence of a vector field defined on radially symmetric domains in the context of vector calculus. We apply straightforward equations of vector calculus with the nabla operator and the transformation matrices from Cartesian to spherical polar coordinates. One needs the divergence of a vector field e.g. to prove that vector fields are eigenfunctions of the Stokes operator on n-dimensional annuli and balls. Our divergence formula in partial derivatives in n-dimensional spherical polar coordinates is an important st","authors_text":"Bern Rummler, Gudrun Th\\\"ater","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AP","submitted_at":"2026-07-01T15:49:42Z","title":"Explicit formulas for gradients and the divergence in n-dimensional spherical coordinates"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.01096","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:0f87002d46cfc24500550b444da6d7bcdc87a9c05568c1d1e2ae5efa718ea469","target":"record","created_at":"2026-07-02T01:18:28Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"38faa5574e83545871d8db35c01af2fbb46aa9923c82e9548ddf7467e79f6210","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AP","submitted_at":"2026-07-01T15:49:42Z","title_canon_sha256":"79cb7bb0616ed87178d36bfdc66a615866b4c8d498aa4e886cbae1afa4025b5d"},"schema_version":"1.0","source":{"id":"2607.01096","kind":"arxiv","version":1}},"canonical_sha256":"69a8ea577efc083623087f580b58ca1619d7a4c838a4b1c0bb7f7efad20b740b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"69a8ea577efc083623087f580b58ca1619d7a4c838a4b1c0bb7f7efad20b740b","first_computed_at":"2026-07-02T01:18:28.926965Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-02T01:18:28.926965Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"69fD/KKmVCuI9W7KQ/XLyoNNKCL1Huli6k9O/U7EjyhbSHaRiWj/JXf6eYB5rePicWxFge2vgsYAJS+gAIK9Dw==","signature_status":"signed_v1","signed_at":"2026-07-02T01:18:28.927366Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.01096","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0f87002d46cfc24500550b444da6d7bcdc87a9c05568c1d1e2ae5efa718ea469","sha256:740a5e90bfc392645771e4c4d707eba1deb42f53a4b554db29ce9c8d853d27bd"],"state_sha256":"08822ee84214f42be6be6700257d390e8a7be3ac6605f5b47773be878c6ca7b1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xLH0qnHr8t23YNxkJ+oCpJU3G7q5g1uQefENMv9T2vN7zT+p+DDN0YnVS54iC8M1ddb2od22byVXG5ySqfw/BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-10T15:11:27.285941Z","bundle_sha256":"1a92f933662c10231c073d764f2ab936fe25945426981b4df09f4c8742799820"}}